ELECTRICITY,* 
SOUND, AND  LIGHT 

MILLIKAN  AND  MILLS 


- 


tr* 


GIFT  OF 
't.P.    Lewis 


A  SHORT  UNIVERSITY  COURSE  IN 

ELECTRICITY,  SOUND,  AND 
LIGHT 


BY 

EGBERT  ANDREWS  MILLIKAN,  PH.D. 

ASSOCIATE  PROFESSOR  OF  PHYSICS  IN  THE  UNIVERSITY  OF  CHICAGO 


JOHN  MILLS,  A.M. 

INSTRUCTOR  IN  PHYSICS  IN  THE  WESTERN  RESERVE  UNIVERSITY 


GINN  &  COMPANY 

BOSTON  •  NEW  YORK  •   CHICAGO  •  LONDON 


$  G  u.  ^ 

M  r-y 

PHYSICS  DEPt. 


PHYSICS  DBPT. 

ENTERED  AT  STATIONERS'  HALL 


COPYRIGHT,  1908,  BY 
ROBERT  A.  MILLIKAN  AND  JOHN  MILLS 


ALL  RIGHTS   RESERVED 

78.9 


GINN   &   COMPANY  •  PRO- 
PRIETORS  •  BOSTON  •  U.S.A. 


PEBFACE 

This  book  represents  primarily  an  attempt  to  secure  a  satisfac- 
tory articulation  of  the  laboratory  and  class-room  phases  of  instruc- 
tion in  physics.  It  is  an  outgrowth  of  the  conviction  that  in  courses 
of  intermediate  grade  in  colleges,  universities,  and  engineering  schools 
a  real  insight  into  the  methods  of  physics,  and  a  thorough  grasp  of 
its  foundation  principles  are  not  readily  gained  unless  theory  is 
presented  in  immediate  connection  with  such  concrete  laboratory 
problems  as  are  calculated  to  give  the  student  a  sound  basis  for 
intelligent  theoretical  work. 

Nevertheless  the  book  is  intended  to  be  much  more  than  a  lab- 
oratory manual.  It  represents  an  attempt  to  present  a  complete 
logical  development,  from  the  standpoint  of  theory  as  well  as  ex- 
periment, of  the  subjects  indicated  in  the  title.  It  is  designed  to 
occupy  a  half  year  of  daily  work,  two  hours  per  day,  in  either  the 
freshman,  sophomore,  or  junior  years  of  the  college  or  technical- 
school  course.  In  the  University  of  Chicago  about  one  half  of  this 
time  is  devoted  to  class  discussions,  lecture-table  demonstrations, 
quizzes,  and  problems,  and  the  remainder  to  laboratory  work.  The 
course  is  preferably  preceded  by  a  similar  course  in  mechanics, 
molecular  physics,  and  heat,  the  two  courses  together  constituting 
a  year's  work  in  college  physics. 

The  method  of  treatment  is  throughout  analytical  rather  than 
descriptive,  although  no  mathematics  beyond  trigonometry  is 
presupposed.  It  is  assumed  that  the  student  has  already  had 
a  beginning  course  in  descriptive  physics  in  the  high  school  or 
elsewhere. 

Most  of  the  apparatus  required  is  of  the  stock  sort  found  in  all 
moderately  well-equipped  college  laboratories.  A  few  special  pieces 
have  been  designed  (Figs.  60,  99,  etc.)  where  for  one  reason  or 
another  existing  forms  seemed  ill  adapted  to  the  needs  of  the 

iii 


€65423 


iv  ELECTRICITY,   SOUND,  AND  LIGHT 

course.  In  the  University  of  Chicago  the  apparatus  is  not,  in  gen- 
eral, used  in  duplicate.  Nine  or  ten  experiments  are  commonly 
kept  going  at  once,  two  pupils  working  together.  The  classes  are 
limited  to  twenty-five. 

The  authors'  thanks  are  due  to  Leeds  &  Northrup  for  the  cut  of 
their  post-office-box  bridge  (Fig.  51  &),  to  Queen  &  Co.  for  the  origi- 
nal of  their  tangent  galvanometer  (Fig.  22),  to  the  American  In- 
strument Company  for  the  original  of  their  voltmeter  (Fig.  40),  and 
to  William  Gaertner  &  Co.  for  the  originals  of  the  magnetometer 
(Fig.  14),  the  voltameter  (Fig.  27),  the  electric  calorimeter  (Fig.  39), 
the  ballistic  galvanometer  (Fig.  60),  the  earth  inductor  (Fig.  107), 
the  ideal  dynamo  (Fig.  99),  and  the  spectrometer  (Fig.  194). 

E.  A.  M. 

UNIVERSITY  OF  CHICAGO  J-  M. 


CONTENTS 


CHAPTER  PAGE 

I.    MAGNETIC  AND  ELECTRIC  FIELDS  OF  FORCE  ....  1 
II.    DETERMINATION    OF    THE    STRENGTHS    OF    MAGNETIC 

FIELDS  AND  MAGNETIC  POLES 17 

III.  MEASUREMENT  OF  ELECTRIC  CURRENTS 28 

IV.  MEASUREMENT  OF  POTENTIAL  DIFFERENCE     ....  46 
V.   MEASUREMENT  OF  RESISTANCE 58 

VI.    TEMPERATURE  COEFFICIENT  OF  RESISTANCE  ;  SPECIFIC 

RESISTANCE 67 

VII.    GALVANOMETER  CONSTANT  OF  A  MOVING-COIL  GALVA- 
NOMETER       77 

VIII.    ABSOLUTE  MEASUREMENT  OF  CAPACITY 90 

IX.    COMPARISON  OF  CAPACITIES,  DETERMINATION  OF  DIE- 
LECTRIC CONSTANTS,  AND  THE  RATIO  OF  THE  ELEC- 
TROSTATIC AND  ELECTRO-MAGNETIC  UNITS  .     .     .  100 
X.    ELECTROMOTIVE  FORCE  AND  INTERNAL  RESISTANCE     .  112 

XL    COMPARISON  OF  ELECTROMOTIVE  FORCES 121 

XII.    ELECTRO-MAGNETIC  INDUCTION 126 

XIII.  CONSTANTS  OF  THE  EARTH'S  MAGNETIC  FIELD    .     .     .  141 

XIV.  SELF-INDUCTION 146 

XV.    MAGNETIC  INDUCTION  IN  IRON 158 

XVI.    ELECTROLYTIC  CONDUCTION 176 

XVII.    VELOCITY  OF  SOUND  IN  AIR 187 

XVIII.    MUSICAL  PROPERTIES  OF  AIR  CHAMBERS 201 

XIX.    LONGITUDINAL  VIBRATIONS  OF  RODS 213 

XX.    WAVES  IN  STRINGS 222 

XXI.    DIFFRACTION  OF  SOUND  AND  LIGHT  WAVES   .     .     .     .  238 

XXII.    DIFFRACTION  GRATING 258 

XXIIL    REFRACTION  OF  LIGHT 270 

XXIV.    TOTAL  REFLECTION 285 

XXV.    PHOTOMETRY 292 

XXVI.    DISPERSION  AND  SPECTRA 302 

XXVII.    POLARIZED  LIGHT 316 

XXVIII.    RADIO-ACTIVITY 333 

PROBLEMS 354 

APPENDIX  OF  TABLES 367 

INDEX 385 

v 


ELECTRICITY,  SOUND,  AKD 
LIGHT 

CHAPTEE  I 
MAGNETIC  AND  ELECTRIC  FIELDS  OF  FORCE 

1.  Quantity  of  magnetism.  It  is  well  known  that  under  cer- 
tain circumstances  bars  of  iron  and  of  some  other  metals,  when 
suspended  so  as  to  be  free  to  rotate  about  a  vertical  axis,  turn  so  as 
to  point  north  and  south.  A  body  which  possesses  the  property 
of  so  doing  is  called  a  magnet.  If  the  two  north-seeking  ends  of 
two  such  magnets  are  brought  near  together,  they  are  found  to 
repel  each  other.  The  same  is  true  of  the  two  south-seeking  ends. 
But  a  north-seeking  and  a  south-seeking  end  are  found  to  attract 
each  other.  On  account  of  these  opposite  characteristics  the  north- 
seeking  ends  of  magnets  are  called  N  poles,  the  south-seeking, 
8  poles.  The  above  facts  may  then  be  stated  in  the  general  law : 
Magnetic  poles  of  like  kind  repel  one  another,  of  unlike  kind 
attract  one  another. 

Different  magnets  of  the  same  length  placed  at  a  given  distance 
from  a  suspended  magnet  are  observed  to  exert  different  forces 
upon  it.  The  quantities  of  magnetism  in  the  poles,  or  the  pole 
strengths,  are  then  arbitrarily  taken  as  proportional  to  the  forces 
exerted.  That  is,  the  force  which  a  given  pole  M  exerts,  at  a 
given  distance,  upon  some  standard  pole  is  taken  as  the  meas- 
ure of  the  number  of  units  of  magnetism  in  M.  It  may  then 
be  proved  experimentally,  by  the  method  used  in  Chapter  II, 
that  the  force  /  exerted  between  any  two  poles  M  and  m  is 
directly  proportional  to  the  product  mM  and  inversely  proportional 


2  :  ELiGTMCLTY,  SOUND,  AND  LIGHT 

to  :the i  ^sqiiar^of  tihei  distance  r  between  them.  The  algebraic 
statement  of  this  experimental  relation  is 

(1) 

in  which  A;  is  a  factor  of  proportionality  depending  upon  the 
choice  of  the  unit  of  magnetism  and  upon  the  medium  through 
which  the  force  acts.  It  has  been  decided  to  choose  this  unit  so 
that  k  equals  unity  for  air ;  that  is,  the  equation  /  =  mM/r2, 
applied  to  air,  contains  the  definition  of  unit  magnetic  pole. 
Thus  if  two  magnets  are  chosen  for  which  m  =  M ,  and  if  r 
is  taken  equal  to  1  cm.,  then  m  is  by  definition  unity  if  /  is 
found  to  be  equal  to  1  dyne.  In  other  words,  a  unit  magnetic 
pole  is  a  pole  of  such  strength  that  when  placed  at  a  distance 
of  1  cm.  from  an  equal  pole  it  repels  or  attracts  it  with  a  force 
of  1  dyne. 

2.  Quantity  of  electricity.  It  is  also  equally  well  known  that 
when  a  glass  rod  has  been  rubbed  with  silk  it  attracts  a  pith  ball, 
but  after  contact  with  the  pith  ball,  repels  it.  Similarly,  when 
ebonite  has  been  rubbed  with  cat's  fur  it  attracts  a  second  pith 
ball,  but  after  contact  with  the  pith  ball,  repels  it.  Furthermore, 
the  pith  ball  which  has  touched  the  rubbed  glass  and  is  repelled 
by  it  is  attracted  by  the  ebonite,  while  the  ball  which  is  repelled  by 
the  ebonite  is  attracted  by  the  glass.  On  account  of  this  behavior 
the  pith  balls  are  said  to  have  been  electrified,  or  to  have  received 
charges  of  electricity  ;  and  on  account  of  the  opposite  characteristics 
of  these  charges  the  one  is  called  positive  and  the  other  negative. 
These  charges  of  electricity  can  be  produced  in  other  ways,  but  in 
every  case  it  has  been  decided  to  call  a  charge  positive  when  it  is 
repelled  by  a  glass  rod  which  has  been  rubbed  with  silk,  negative 
when  it  is  repelled  by  an  ebonite  rod  which  has  been  rubbed  with 
cat's  fur.  The  above  facts  may  then  be  stated  in  the  general  law : 
Electrical  charges  of  like  sign  repel  one  another,  of  unlike  sign 
attract  one  another.  It  need  scarcely  be  said  that  in  adopting 
these  conventions  and  in  setting  up  this  law  no  assumption 
whatever  has  been  made  regarding  the  nature  of  electricity. 
It  has  merely  been  agreed  to  call  a  body  electrified,  or  charged 


MAGNETIC  AND  ELECTRIC  FIELDS  OF  FORCE        3 

with  electricity,  which  behaves  toward  pith  balls  or  other  objects 
as  does  the  rubbed  glass  or  the  rubbed  ebonite. 

Definitions  precisely  similar  to  those  used  in  the  quantitative 
study  of  magnetism  are  adopted  also  in  the  quantitative  study  of 
electricity.  Thus  if  q  and  Q  are  two  electric  charges  the  magni- 
tudes of  which  are  measured  by  the  forces  which  they  exert  upon 
a  third  charge  at  a  given  distance  from  it,  then  it  can  be  proved 
experimentally  that 


As  in  magnetism,  so  in  electricity,  the  unit  of  quantity  is  chosen 
so  that  for  action  between  charges  separated  by  air 

/=£<?.  (2) 

r2 

This  equation  contains,  then,  the  definition  of  unit  charge.  In 
words,  unit  quantity  of  electricity  (unit  charge)  is  defined  as  that 
quantity  which  placed  in  air  at  a  distance  of  1  cm.  from  an  equal 
quantity  acts  upon  it  with  a  force  of  1  dyne. 

3.  Electrical  conduction.    If  a  charged  ebonite  rod  is  rubbed 
over  one  end  of  a  long  metal  body  which  rests  upon  sealing  wax 
or  glass,  a  pith  ball  placed  near  the  remote  end  of  the  metal  body 
will  at  once  be  attracted  to  it.    If  the  metal  body  is  replaced  by 
one  of  glass,  or  wood,  or  almost  any  nonmetallic  solid,  no  effect 
whatever  is  produced  upon  the  pith  ball.    In  view  of  experiments 
of  this  sort,  it  is  customary  to  divide  substances  into  two  classes, 
conductors  and  insulators,  or  nonconductors,  according   to   their 
ability  to  transmit  electrical  charges.    Thus  metals  and  solutions 
of  salts  and  acids  in  water  are  all  conductors  of  electricity,  while 
porcelain,  rubber,  mica,  shellac,  wood,  silk,  vaseline,  turpentine, 
paraffin,  and  oils  generally  are  insulators.    No  hard  and  fast  line, 
however,  can  be  drawn  between  conductors  and  insulators,  since 
substances  can  be  found  of  all  degrees  of  conductivity  between 
that  of  sulphur,  amber,  or  quartz,  the  best  insulators,  and  that 
of  silver  and  copper,  the  best  conductors. 

4.  Distinctions  between  electricity  and  magnetism.    The  fact 
of  conduction  constitutes  one  of  the  most  essential  distinctions 
between  electricity  and  magnetism.    Electrically  charged  bodies 


4  ELECTRICITY,  SOUND,  AND  LIGHT 

lose  their  charges,  in  part  at  least,  as  soon  as  they  are  touched  by 
conductors,  but  such  treatment  has  no  influence  whatever  upon 
magnetic  poles.  The  phenomena  of  magnetism  therefore  show 
nothing  which  is  at  all  analogous  to  the  phenomenon  of  con- 
duction in  electricity.  Furthermore,  all  bodies,  conducting  or 
nonconducting,  can  be  strongly  electrified  by  friction  if  they  are 
mounted  upon  insulating  supports,  but  only  iron,  steel,  nickel, 
and  some  newly  discovered  alloys  of  copper,  magnesium,  and  alu- 
minum, called  Heussler  alloys,  can  be  appreciably  magnetized. 
Magnetism  and  electricity  are  then  to  be  regarded  as  distinct 
phenomena.  A  peculiar  relationship,  however,  which  has  been 
found  to  exist  between  them  will  be  discussed  in  Chapter  III. 

5.  Electrostatic  induction.  If  a  positively  charged  body  A 
(Fig.  1)  is  placed  in  the  neighborhood  of  an  uncharged  body  B, 
which  is  supplied  with  pith  balls  or  strips  of  paper  a,  b,  c,  as 

shown  in  the  figure, 
the  divergence  of  a 
and  c  will  show  that 
the  ends  of  B  have 
received  electrical 

v ^      charges,   while   the 

failure  of  b  to  di- 
verge will  show  that  the  middle  of  B  is  uncharged.  Further,  a 
positively  charged  glass  rod  will  be  found  to  repel  c  and  attract  a. 
The  experiment  illustrates  the  fact  that  the  mere  influence  which 
an  electric  charge  exerts  upon  a  conductor  placed  in  its  neighbor- 
hood is  able  to  produce  electrification  in  that  conductor,  the  remote, 
end  receiving  a  charge  of  sign  like  that  of  the  original  charge,  while 
the  near  end  has  a  charge  of  opposite  sign.  If  A  is  removed  the 
charges  at  a  and  b  entirely  disappear,  and  the  conductor  B  is  found 
to  be  altogether  uncharged,  thus  showing  that  the  total  amount  of 
positive  electricity  which  appeared  at  one  end  of  B  must  have  been 
exactly  equal  to  the  total  amount  of  negative  which  appeared  at 
the  other  end.  The  phenomenon  of  the  appearance  of  equal  and 
opposite  electrical  charges  in  the  opposite  ends  of  a  conductor  placed 
near  a  charged  body  is  known  as  electrostatic  induction,  and  a  con- 
ductor in  this  condition  is  said  to  be  electrically  polarized. 


MAGNETIC  AND  ELECTRIC  FIELDS  OF  FOECE 


6.  Positive  and  negative  electricities  always  appear  in  equal 
amount.    That  positive  and  negative  electricities  always  appear  in 
exactly  equal  amount,  as  well  when  the  electrification  is  produced 
by  friction  as  by  induction,  may  be  convincingly  shown  by  attach- 
ing a  piece  of  fur  or  flannel  to  the  end  of  a  strip  of  ebonite,  rub- 
bing with  it  the  end  of  another  similar  strip,  bringing  the  two 
together,  without  separating  them,  near  a  charged  pith  ball  or 
other  electroscope,  then  separating  them  and  bringing  each  in  suc- 
cession near  the  electroscope.     So  long  as  they  are  together  they 
will  exhibit  no  electrification  whatever,  but  when  separated  they 
will  show  charges  of  opposite  sign.    That  these  charges  are  exactly 
equal  is  shown  by  the  fact  that  they  exactly  neutralized  each  other 
before  the  separation. 

The  test  for  the  equality  of  the  two  charges  may  be  made  ex- 
tremely delicate  by  inserting  the  two  rubbed  bodies  together  into 
a  hollow  metal  vessel  to  which  a  gold-leaf  electroscope  is  connected, 
as  in  Figure  2.  So  long 
as  the  two  rubbed  bodies 
are  together  the  leaf  will 
show  no  trace  of  diver- 
gence, but  when  one  of 
them  is  removed  the  leaf 
will  stand  out  in  the 
position  of  the  dotted  line. 
When  this  rubbed  body 
is  replaced  by  the  other 
the  divergence  will  be  of 
equal  amount,  but  the  charge  will  be  of  opposite  sign,  as  can  be 
shown  by  bringing  a  positively  charged  glass  rod  near  the 
electroscope ;  for  then,  if  the  latter  already  had  a  positive  charge, 
the  divergence  will  be  increased  by  the  approach  of  the  glass  rod, 
but  if  it  had  a  negative  charge  the  divergence  will  be  diminished. 

7.  Theories  of  electricity.    During  the  nineteenth  century  the 
facts  of  electricity  were  most  commonly  described  in  terms  of  the 
so-called   two-fluid  theory.     Although    this    theory   is  no  longer 
regarded  as  corresponding  closely  to  reality,  and  although  it  has 
perhaps  never  been  generally  accepted  as  anything  more  than  a 


FIG.  2 


6  ELECTRICITY,  SOUND,  AND  LIGHT 

convenient  fiction,  it  is  so  intimately  related  to  the  present  nomen- 
clature of  the  subject  as  to  deserve  careful  attention.  According 
to  it  all  bodies  contain  equal  amounts  of  two  weightless  electrical 
fluids,  positive  electricity  and  negative  electricity.  These  fluids  are 
self-repellent  but  mutually  attractive,  so  that  their  effects  com- 
pletely neutralize  one  another  in  bodies  in  the  normal  condition. 
But  if  a  conductor,  i.e.  a  body  in  which  the  fluids  are  able  to  move 
about,  is  brought  near  a  charged  body,  the  fluid  of  like  sign  is 
driven  to  the  remote  end,  while  the  fluid  of  unlike  sign  is  drawn 
to  the  near  end  of  the  conductor.  This  furnishes  the  explanation 
of  the  phenomenon  of  induction. 

In  insulators  the  fluids  are  supposed  not  to  be  free  to  move 
from  point  to  point,  but  friction  between  dissimilar  substances 
causes  electrical  separation  in  the  adjacent  boundary  layers  of  the 
dissimilar  substances,  the  excess  of  positive  which  goes  to  one 
body  being  necessarily  equal  to  the  deficiency  of  positive,  i.e.  the 
excess  of  negative,  which  is  left  upon  the  other. 

The  modern  modification  of  the  two-fluid  hypothesis  is  that 
which  has  been  developed  within  the  last  decade  by  Drude  and 
Riecke  in  Germany.  It  is  identical  with  the  older  two-fluid 
theory,  save  that  it  replaces  the  weightless  and  continuous  elec- 
trical fluids  by  equal  numbers  of  positive  and  negative  corpuscles, 
or  electrons,  which  are  assumed  to  be  constituents  of  the  atoms  of 
all  substances,  but  which,  in  the  case  of  conductors,  are  continually 
becoming  detached  from  the  atoms ;  so  that  at  a  given  instant 
there  are  always  present  free  positive  and  free  negative  corpuscles 
which  are  able  to  move  through  the  conductor  in  opposite  direc- 
tions under  the  influence  of  any  outside  electrical  force.  Con- 
ductors differ  from  insulators  only  in  that  the  atoms  of  the  latter 
do  not  lose  their  corpuscles  to  any  appreciable  extent. 

The  old-time  rival  of  the  two-fluid  theory  was  the  so-called  one- 
fluid  theory,  originally  due  to  Benjamin  Franklin.  It  differed  from 
the  two-fluid  theory  only  in  regarding  a  positive  charge  as  indicat- 
ing an  excess,  a  negative  charge,  a  deficiency  in  a  certain  normal 
amount  of  one  single,  all-pervading  electrical  fluid,  viz.  positive 
electricity,  which  was  self-repellent  but  strongly  attracted  by  ordi- 
nary matter.  In  order  to  account  for  the  mutual  repulsions  of 


MAGNETIC  AND  ELECTRIC  FIELDS  OF  FOKCE        7 

negatively  charged  bodies,  it  was  found  necessary  to  assume  that 
the  particles  of  ordinary  matter,  when  dissociated  from  electricity, 
repelled  one  another. 

A  modern  modification  of  the  one-fluid  theory  has  recently 
come  into  prominence  through  the  combined  work  of  several 
physicists  in  high  standing,  notably  Lord  Kelvin  and  J.  J. 
Thomson.  According  to  this  theory  a  certain  amount  of  posi- 
tive electricity  is  supposed  to  constitute  the  nucleus  of  the  atom 
of  every  substance.  About  the  center  of  this  positive  charge  are 
grouped  a  number  of  very  minute  negatively  charged  corpuscles, 
or  electrons,  the  mass  of  each  of  which  is  approximately  2-$-$-$  of 
that  of  the  hydrogen  atom.  The  sum  of  the  negative  charges 
of  these  electrons  is  supposed  to  be  just  equal  to  the  positive 
charge  of  the  atom,  so  that  in  its  normal  condition  the  whole 
atom  is  neutral  or  uncharged.  But  in  the  jostlings  of  the  mole- 
cules of  a  conductor  electrons  are  continually  becoming  detached 
from  the  atoms,  moving  about  freely  between  the  molecules,  and 
then  reentering  other  atoms  which  have  lost  electrons.  Therefore, 
at  any  given  instant,  there  are  always  present  in  any  conductor 
a  large  number  of  free  negative  electrons  and  an  exactly  equal 
number  of  atoms  which  have  lost  electrons,  and  which  are  there- 
fore positively  charged.  Such  a  conductor  would,  as  a  whole, 
show  no  charge  either  of  positive  or  of  negative  electricity.  But 
the  presence  near  it  of  a  body  charged,  for  example,  negatively 
would  cause  the  negatively  charged  electrons  to  stream  away  to 
the  remote  end,  leaving  behind  them  the  positively  charged  atoms, 
which,  in  solids,  are  not  supposed  to  be  free  to  move  appreciably 
from  their  positions.  In  the  presence  of  a  positively  charged  body, 
on  the  other  hand,  the  electrons  would  be  attracted  to  the  near 
end,  while  the  remote  end  would  be  left  with  the  immovable 
positive  atoms. 

The  only  advantage  of  this  theory  over  that  which  assumes  the 
existence  of  two  types  of  corpuscles  is  that,  while  there  is  much 
direct  experimental  evidence  for  the  existence  of  negative  cor- 
puscles of  about  2-$-Q-Q  the  mass  of  the  hydrogen  atom,  no  direct 
evidence  whatever  for  the  existence  of  such  positively  charged 
corpuscles  has  as  yet  been  brought  to  light.  In  general,  wherever 


8  ELECTRICITY,  SOUND,  AND  LIGHT 

positively  charged  bodies  appear  they  are  found  to  be  of  atomic  size. 
The  negative  corpuscles,  on  the  other  hand,  are  sometimes  found 
as  constituents  of  atoms,  sometimes  as  independent  detached  bodies. 

It  will  be  seen  that  this  last  theory  is  like  Franklin's  in  that  it 
assumes  but  one  movable  kind  of  electrical  matter,  i.e.  one  electrical 
fluid,  while  it  is  unlike  it  in  making  this  fluid  negative  instead  of 
positive,  and  also  in  making  it  consist  of  discrete  particles.  It  is  like 
the  two-fluid  theory,  however,  in  postulating  the  existence  of  two 
distinct  entities  called  respectively  positive  and  negative  electricity. 
The  positive  electricity,  however,  plays  quite  the  same  rc>le  which 
in  the  old  one-fluid  theory  was  assigned  to  ordinary  matter. 

8.  Fields  of  force.  Afield  of  force  is  simply  a  region  in  which 
force  exists.  It  may  be  a  magnetic,  electrical,  or  gravitational  field 
which  is  under  consideration.  The  strength  of  field  at  any  point 
in  such  a  region  is  the  number  of  units  of  force  which  unit  quan- 
tity (be  it  mass,  pole  strength,  or  charge)  experiences  at  the  point 
considered.  Thus  the  strength  of  field  at  a  point  1  cm.  distant 
from  a  unit  pole  (conceived  as  concentrated  at  a  point)  is  unity. 
Unit  field  is,  then,  a  field  in  which  unit  quantity  experiences  1  dyne 
of  force.  For  example,  the  strength  of  magnetic  field  at  a  given 
point  in  space  is  ten  units  if  unit  pole  experiences  10  dynes  of 
force  when  placed  at  this  point. 

The  direction  of  a  gravitational  field  at  any  point  is  defined  as 
the  direction  in  which  a  small  quantity  of  matter  would  tend  to 
move  if  placed  in  the  field  at  the  point  considered.  The  direction 
of  a  magnetic  field  is  defined  as  the  direction  in  which  an  isolated 
N  pole  would  move.  The  direction  of  an  electric  field  is  defined 
as  the  direction  in  which  an  isolated  positive  charge  of  electricity 
would  move. 

A  line  of  force  in  any  one  of  these  fields  is  the  direction  in 
which  a  free  mass,  a  free  N  pole,  or  a  free  positive  charge  would 
move  if  it  had  no  inertia.  It  is  convenient  and  customary,  how- 
ever, to  conceive  of  as  many  lines  drawn  across  any  square  centi- 
meter taken  at  right  angles  to  the  direction  of  the  force  as  the 
field  possesses  units  of  strength  at  the  point  considered.  The  line 
of  force  then  becomes  the  unit  of  field  intensity  or  strength ;  that 
is,  in  a  gravitational  field  a  line  means  a  field  strength  such  that 


MAGNETIC  AND  ELECTRIC  FIELDS  OF  FORCE        9 

a  force  of  1  dyne  acts  on  every  gram  placed  in  it.  Thus,  since 
the  earth's  field  strength  at  the  surface  is  980  dynes,  it  is  cus- 
tomary to  consider  980  lines  of  gravitational  force  as  piercing 
each  square  centimeter  of  the  earth's  surface.  In  magnetic  fields, 
in  connection  with  which  the  convention  is  most  commonly  used, 
a  line  means  a  field  strength  such  that  a  force  of  1  dyne  acts  on 
each  unit  pole.  It  has  received  the  special  name  of  a  gauss.  In 
electrical  fields  the  more  usual  term  is  tube  of  force,  and  the  con- 
ception is  that  as  many  tubes  of  force  cross  any  square  centimeter 
at  right  angles  to  the  direction  of  the  field  as  there  are  units  in 
the  field  strength  at  that  point. 

With  these  conventions  it  is  evident  that  a  uniform  field  is 
represented  by  a  system  of  parallel  lines  of  force,  and  conversely, 
that  where  the  lines  of  force  are  parallel,  the  field  has  everywhere 
the  same  strength.  A  convergent  system  of  lines  represents  a 
field  of  increasing  strength;  a  divergent  system  of  lines  repre- 
sents a  field  of  decreasing  strength.  It  need  scarcely  be  said  that 
a  line  of  force  has  no  objective  reality.  The  representation  of 
fields  of  force  by  lines  is  a  matter  of  convenience  only. 

9.  Gravitational  potential.    The   term   "potential"  was  first 
used  in  connection  with  gravitational  forces.    Potential  is  a  char- 
acteristic of  a  point .  in  space,  not,  in  general,  of  .  tf 
a  body.    It  may  be  looked  upon  merely  as  an 
abbreviation  for   the    expression   "the   potential 
energy   of   unit   mass   at  the   point   considered." 
Thus  at  a  point  a  above  the  surface  of  the  earth 
(Fig.  3)  a  gram  of  mass  possesses  a  certain  po- 
tential energy  with  reference  to  a  point  6  on  the 
surface  of  the  earth.    This  potential  energy  is  the 
amount  of  work  that  the  gram  of  mass  can  do  in 
f ailing  to  the  earth.    Since  the  term  "  potential "  is 
merely  an  abbreviation  for  the  potential  energy  of  unit  mass,  ic  is 
evident  that  potential  is  measured  in  energy  or  work  units.    The 
potential  of  a  point  is  unity  when  it  requires  one  erg  of  work  to 
bring  unit  mass  from  the  point  which  is  taken  as  the  zero  of 
potential  up  to  the  point  considered.    The  potential  energy  of  unit 
mass  at  the  point  a  (see  Fig.  3)  is  greater  if  the  point  c,  below  the 


10  ELECTRICITY,  SOUND,  AND  LIGHT 

surface  of  the  earth,  is  taken  as  the  point  of  reference  instead  of 
the  point  b.  Thus  it  is  evident  that  some  point  must  be  chosen 
arbitrarily  from  which  to  reckon  the  potential  of  other  points. 

A  point  which  is  infinitely  distant  from  all  attracting  or  repelling 
bodies  is  taken  as  the  point  from  which  absolute  potential  is  reckoned, 
i.e.  such  a  point  is  taken  as  the  absolute  zero  of  potential.  The  abso- 
lute potential,  then,  of  any  point  in  the  universe  is  the  number  of 
ergs  of  work  which  must  be  done  (by  some  outside  agent)  to  bring 
unit  mass  from  this  absolute  zero  up  to  the  point  considered. 

In  general,  we  are  more  concerned  with  the  difference  in  potential 
(usually  written  P.D.)  between  two  points  than  with  the  absolute 
potentials  of  the  points.  The  P.D.  between  two  points  is  then  defined 
as  the  amount  of  work  which  the  external  agent  must  do  in  order 
to  carry  the  unit  mass  from  the  point  of  the  lower  to  that  of  the 
higher  potential ;  or,  what  amounts  to  the  same  thing,  the  P.D,  is 
the  amount  of  work  which  the  acting  force  does  in  carrying  unit 
mass  from  the  point  of  higher  to  that  of  lower  potential. 

Now  the  force  of  gravitation  is  always  an  attractive,  never  a 
repellent,  force.  For  all  points  which  are  at  a  finite  distance 
from  any  astronomical  body  the  work  which  the  external  agent 
must  do  in  bringing  unit  mass  from  an  infinite  distance  to  any 
point  is  therefore  less  than  nothing.  That  is,  the  work  is  done 
not  by,  but  against,  the  action  of  the  external  agent.  From  the 
definition  of  absolute  potential,  as  given  above,  it  is  evident  that 
the  gravitational  potential  of  all  points  within  a  finite  distance 
of  any  astronomical  body  must  be  negative. 

10.  Magnetic  potential.  The  above  definitions  hold  almost 
without  change  for  magnetic  forces,  save  that  it  is  necessary  to 
specify  whether  the  unit  quantity  is  an  N  pole  or  an  8  pole. 
The  magnetic  potential  of  a  point  is  defined  as  the  amount  of  work 
which  an  external  agent  must  do  against  the  existing  magnetic 
field  in  order  to  bring  a  unit  N  pole  from  infinity  up  to  the  point 
considered.  Thus  the  potential  in  the  immediate  neighborhood  of 
an  N  pole  is  evidently  positive,  because  the  unit  N  pole  is 
repelled,  and  hence  the  external  agent  must  do  work  in  order  to 
bring  up  the  pole  from  infinity.  It  is  equally  evident  that  the 
potential  in  the  immediate  neighborhood  of  an  8  pole  is  negative. 


MAGNETIC  AND  ELECTRIC  FIELDS  OF  FORCE      11 

11.  Electrical  potential.    Similarly,  the  electrical  potential  of  a 
point  is  the  amount  of  work  required  to  bring  unit  positive  charge 
from  infinity  up  to  the  point  considered.    Points  in  the  neighbor- 
hood   of  a  positive    charge  have  therefore  a  positive  potential ; 
those  in  the  neighborhood  of  a  negative  charge  have  a  negative 
potential.    These  definitions  apply  as  completely  in  the  study  of 
so-called  current  electricity  as  in  that  of  static  electricity.     Under 
all  circumstances  the  term  P.D.  means  the  amount  of  work  required 
to  carry  unit  positive  charge  between  the  two  points  considered. 

12.  Equipotential  surfaces.    An   equipotential  surface  is  the 
locus  of  a  system  of  points  all  of  which  have  the  same  potential. 
Thus  the  gravitational  equipotential  surfaces  about  the  earth  are 
approximately  spherical  surfaces  concentric  with  it,  because  the 
amount  of  work  required  to  bring  unit  mass  to  within  a  certain 
distance  of  the  center  of  the  earth  is  the  same,  no  matter  from 
what  side  the  earth  is  approached. 

Now  it  can  be  shown  that  the  direction  of  the 
field  of  force  at  any  point  is  perpendicular  to  the 
equipotential  surface  passing  through  that  point. 
In  order  to  prove  this  statement  it  is  first  necessary 
to  show  that  the  work  done  hi  carrying  a  body 
between  any  two  points  is  independent  of  the 
path  chosen.  If  a  body  is  carried  from  b  to  a  over 
the  path  Ida  (Fig.  4)  a  certain  amount  of  work  w 
is  done  upon  it.  Now  suppose  the  work  of  the 
path  Ida  is  less  than  that  of  the  path  lea.  Then 
when  the  body  returns  to  b  over  the  path  acb  it  will  give  up  a 
certain  amount  of  energy  w' ,  i.e.  it  will  do  a  certain  amount  of 
work  w'.  As  a  net  result  of  the  operation  we  have  expended  an 
amount  of  work  w  and  have  received  back  a  larger  amount  of 
work  w',  and  yet  have  brought  everything  back  to  the  initial 
condition.  We  have  therefore  created  an  amount  of  energy 
w1 — w.  But  according  to  the  doctrine  of  the  conservation  of 
energy  this  is  impossible.  Hence  w'  cannot  be  greater  than  w. 
By  reversing  the  operation  it  can  be  proved  that  w  cannot  be 
greater  than  w'.  That  is,  the  work  of  the  path  Ida  is  equal  to 
that  of  any  other  path  lea. 


12  ELECTRICITY,  SOUND,  AND  LIGHT 

Now  let  nop  (Fig.  5)  represent  any  equipotential  surface.    By 
the  definition  of  such  a  surface  the  work  required  to  move  a  unit 
body  from  any  arbitrary  zero,  say  m,  over  the  distance  mn  is  the 
same  as  that  required  to  move  it  over  the  dis- 
tance mo.     But  by  the  proposition  just  proved 
the  work  of  the  path  mn  is  equal  to  the  work 
of  the  path  mo  plus  on.    Hence  the  work  corre- 
FIG  5  spending   to   the   path  on  must   be  zero.     If   it 

requires  no  work  to  move  a  body  over  an  equi- 
potential surface,  the  existing  force  can  have  no  component  along 
that  surface,  i.e.  the  force  must  be  everywhere  normal  to  an 
equipotential  surface. 

13.  Potential  of   a  conductor  in  electrical  equilibrium.    That 

the  electrical  potentials  of  all  points  on  or  within  a  conductor  in 

the  static  condition  must  be  the  same  follows  at  once  from  the 

fact  of   conductivity.    For  as  soon  as  a  conductor  ab  (Fig.  6)  is 

brought  into  the  field  of  a  positively  charged  body  c  the  negative 

electricity  within  ab  at  once  moves  toward  b  and  a  positive  charge 

appears  at  a  until  further  movement  of  negative  toward  b  is  checked 

by  the  action  of  the  negative  accumulated  at  b  and  the  positive 

.    accumulated  at  a.     It  is  obvious,  then,  that  there  can  be  electrical 

\  equilibrium  within  a  conductor 

\only   when   all   electrical  forces  _     C _  _j_   j 

within  the  conductor  have  been  /"*/    ^- 

reduced  to  zero.  No  electrical 
force,  then,  can  exist  within  a 
conductor  in  the  static  con-  ,  ,  FIG  6 

ditioii;  hence  no  work  can  be 
required  to  move  unit  charge  from  one  point  to  another  within  the 
conductor.  It  follows  that  all  points  within  or  upon  a  conductor 
in  the  static  condition  must  have  the  same  potential.  The  above 
reasoning  holds  as  well  for  hollow  as  for  solid  conductors,  provided 
no  insulated  charged  bodies  exist  within  the  hollow  portion. 

The  experimental  verification  of  these  conclusions  was  first 
made  by  Faraday,  who  covered  a  large  box  with  tin  foil  and  went 
inside  with  very  delicate  electroscopes.  These  remained  wholly 
unaffected  even  when  powerful  electrical  disturbances  took  place 


MAGNETIC  AND  ELECTRIC  FIELDS  OF  FORCE      13 


just  outside  the  box.  In  laboratory  practice  it  is  now  customary 
to  screen  delicate  electrostatic  instruments  from  external  disturb- 
ances by  surrounding  them  with  sheet  metal  or  wire  gauze. 

Since,  then,  all  points  on  a  conductor  have  the  same  potential, 
it  has  become  customary  to  speak  of  the  potential  of  the  con- 
ductor rather  than  of  the  potential  of  points  on  the  conductor,  in 
spite  of  the  fact  that  the  term  "potential"  is  one  which  in  strictness 
characterizes  points  in  space  rather  than  bodies.  Further,  since 
the  surface  of  a  conductor  in  electrical  equilibrium  is  always  an 
equipotential  surface,  it  follows  that  electrical  lines  of  force  always 
enter  or  leave  a  conductor  normally  to  the  surface. 

14.  Mapping  equipotential  surf  aces.  It  is  evident  from  section  12 
that  if  it  is  possible  to  map  the  direction  of  the  lines  of  force  in 
any  field,  it  must  also  be  possible  to  map  out  the  equipotential 
surfaces  in  that  field.  Thus,  suppose  it  to  be  required  to  find  the 
intersections  with  the  plane  of  the  paper  of  the  equipotential  sur- 
faces about  two  isolated  spheres  whose  centers  are  in  the  plane  of 


FIG.  7 

the  paper.  A  little  consideration  of  the  field  of  force  about  two  such 
spheres  will  show  that  the  general  shape  of  the  lines  of  force  is  that 
represented  by  the  full  lines  of  the  sketch  (Fig.  7).  By  drawing 
lines  everywhere  perpendicular  to  these  force  lines  the  equipotential 
lines  represented  by  the  dotted  lines  of  the  figure  are  obtained. 


14 


ELECTRICITY,  SOUND,  AND  LIGHT 


EXPERIMENT  1 

(A)  Object.  To  map  completely  a  somewhat  complicated  magnetic  field, 
and  to  draw  the  equipotential  lines. 

Directions.  The  field  here  plotted  will  be  the  combination  of  the  field 
of  the  earth  and  the  fields  of  two  magnets.  Two  straight  bar  magnets 
(about  one  foot  long)  are  to  be  placed  vertically  in  a  box  the  top  of  which 
(about  three  feet  square)  is  shown  in  Figure  8.  The  magnets  pass  through 
the  holes  a  and  b  and  their  upper  ends  are  flush  with  the  surface  of  the  box. 
First  see  that  the  line  connecting  a  and  b  is  in  the  magnetic  meridian  ; 
then,  for  the  first  exercise,  place  at  a  a  north  pole  and  at  &  a  south  pole. 

With  brass  pins  attach  to 
the  box  a  sheet  of  paper 
large  enough  to  cover  the 
entire  top. 

The  direction  in  which 
the  north  pole  of  a  short 
compass  needle  points 
when  the  compass  is 
placed  in  a  magnetic  field 
represents  the  direction 
of  the  magnetic  force  at 
the  center  of  the  compass 
needle  ;  for  the  two  poles 
are  always  under  the' 
action  of  forces  which 
tend  to  move  them  in 
opposite  directions  along 
the  line  connecting  them, 
and  which  would  so  move 
them  if  they  could  be  de- 
tached from  one  another. 

Hence  the  following  method  of  plotting  the  direction  of  the  magnetic 
lines  at  once  suggests  itself.  Set  the  needle  at  some  point  in  the  field 
and  with  a  sharp  pencil  place  two  dots,  e.g.  1  and  2  (Fig.  8)  directly 
under  the  extremities  of  the  needle.  Move  the  needle  along  until  the 
pole  which  stood  over  1  now  stands  directly  over  2,  and  make  a  new  dot  S 
to  indicate  the  position  of  the  other  pole.  Proceed  thus  until  the  line 
either  runs  into  a  pole  or  leaves  the  paper.  In  this  way  plot  enough 
lines  to  show  a  complete  outline  of  the  field.  Plot  with  especial  care 
the  lines  which  pass  near  the  singular  points,  i.e.  points  in  which  the 
needle  will  assume  no  definite  position. 

As  a  second  exercise  plot  the  field  produced  when  both  a  and  b  are  north 
poles.  When  the  outline  of  these  fields  has  been  obtained  estimate  the 


FIG.  8 


MAGNETIC  AND  ELECTRIC  FIELDS  OF  FOECE      15 

directions  of  the  intermediate  lines  wherever  it  is  perfectly  evident  what 
their  general  course  must  be,  and  thus  fill  in  the  field  thickly  with  force 
lines.  Then  upon  each  of  the  two  fields  draw,  in  red  ink,  a. second  system 
of  lines  which  is  everywhere  roughly  perpendicular  to  the  first  system, 
-.i.e.  draw  the  equipotential  lines. 

(B)  Object.  To  map  the  equipotential  surfaces  of  an  electrical  field,  and 
to  draw  the  force  lines. 

Directions.  In  Figure  9  is  shown  a  tray  on  the  glass  bottom  of  which  is 
pasted  a  sheet  of  coordinate  paper.  The  tray  is  filled  to  a  depth  of  three 
or  four  millimeters  with  a  solution  of  ammonium  chloride  or  of  any  con- 
venient salt.  A  and  B  are  the  two  points  which  are  to  be  maintained  at 
different  potentials  and  about  which  it  is  desired  to  find  the  equipotential 
lines  which  lie  in  the  plane  of  the  liquid.  If  both  of  the  terminals  of  an 


FIG.  9 


instrument  for  detecting  an  electrical  current  lie  on  the  same  equipotential 
surface,  no  current  passes  through  the  instrument.  If,  however,  the  two 
terminals  lie  on  surfaces  of  different  potentials,  a  charge  is  urged  through 
the  instrument  from  the  point  of  higher  to  that  of  lower  potential,  and 
the  instrument  indicates  the  fact.  If  one  terminal  be  fixed  and  the  other 
moved  to  various  points  of  the  field,  the  equipotential  surface  on  which 
the  first  terminal  lies  will  pass  through  all  the  positions  of  the  second  for 
which  no  current  flows  through  the  instrument.  The  telephone  T  is  used 
for  detecting  a  difference  of  potential  between  its  terminals  C  and  D. 


16  ELECTRICITY,  SOUND,  AND  LIGHT 

Since  the  telephone  receiver  will  respond  by  buzzing  only  as  the  result  of 
an  intermittent  or  alternating  difference  of  potential,  it  is  necessary  that 
A  and  B  be  maintained  at  a  constantly  varying  P.D.  But  whatever  the 
P.D.  between  A  and  B  the  equipotential  lines  will  preserve  the  same  con- 
figuration, although,  of  course,  assuming  different  absolute  values.  This 
varying  P.D.  is  obtained  by  connecting  A  and  B  to  the  secondary  S  of  a 
small  induction  coil,  to  the  primary  P  of  which  is  connected  a  storage 
battery  or  dry  cell  E. 

Assume  two  lines  on  the  coordinate  paper  as  axes  and  locate  points 
from  these.  Connect  A  and  B  as  described  and  set  them  at  two  points 
about  20  cm.  apart.  Set  the  electrode  C  about  1  cm.  from  A,  toward  B. 
With  the  electrode  D  locate  enough  points  for  which  there  is  no  buzzing 
in  the  receiver  to  plot  the  equipotential  line  through  the  point  at  which 
C  is  placed.  Move  C  to  a  position  about  one  sixth  of  the  remaining  dis- 
tance toward  B  and  repeat.  Continue,  moving  C  another  sixth,  until  the 
whole  field  has  been  explored.  Plot  on  coordinate  paper  to  one  half  scale 
the  lines  so  found.  The  lines  of  force  will  form  a  system  everywhere  at 
right  angles  to  these  equipotential  lines.  Draw  this  system  in  red  ink. 


CHAPTEK   II 

THE  DETERMINATION  OF   THE  STRENGTHS  OF  MAGNETIC 
FIELDS  AND  OF  MAGNETIC  POLES 

15.  Every  magnet  possesses  equal  amounts  of  N  and  S 
magnetism.  The  preceding  chapter  dealt  only  qualitatively  with 
magnetic  and  electrical  phenomena.  The  present  chapter  has  to 
do  with  the  exact  measurement  of  magnetic  quantities.  One  of 
the  most  important  of  these  quantities  is  the  so-called  magnetic 
moment  of  a  magnet.  In  order  to  gain  a  clear  conception  of  the 
meaning  of  this  term  it  is  first  desirable  to  consider  why  we  believe 
that  the  N  and  S  poles  of  any  particular  magnet  are  of  exactly 
equal  strength. 

The  earth  behaves  like  a  huge  magnet,  one  pole  of  which  is 
situated  near  the  north  geographical  pole,  the  other  near  the  south 
geographical  pole.  According  to  our  convention  the  former  of 
these  is  an  $  pole,  the  latter  an  N  pole.  Since  both  of  the  poles 
of  the  earth  are  very  remote  in  comparison  with  the  length  of 
any  ordinary  magnet,  it  follows  that  if  the  north  and  south  poles 
of  a  suspended  magnet  are  of  equal  strength,  then  the  attraction 
which  either  of  the  earth's  poles  exerts  upon  one  of  the  ends  of 
the  suspended  magnet  cannot  be  sensibly  different  from  the  repul- 
sion which  this  same  pole  exerts  upon  the  other  end  of  the  magnet. 
Conversely,  if  the  attraction  which  the  northern  pole  of  the  earth 
exerts  upon  the  N  pole  of  a  magnet  is  found  to  be  equal  to  the 
repulsion  which  it  exerts  upon  the  S  pole,  then  these  two  poles 
must  be  of  equal  strength.  A  similar  method  of  reasoning  may  be 
applied  to  the  action  of  the  southern  pole  of  the  earth.  Now 
experiment  shows  that  a  floating  magnet  experiences  no  motion 
of  translation  toward  north  or  south,  but  only  a  motion  of  rotation 
about  a  vertical  axis.  Since  this  experiment  may  be  performed 
with  any  magnet,  it  follows  that  the  N  and  S  poles  of  a  magnet 
are  always  of  equal  strength. 

17 


18 


ELECTRICITY,  SOUND,  AND  LIGHT 


m' 


16.  Magnetic  moment.  The  moment  of  force  tending  to  pro- 
duce rotation  in  a  magnet  placed  at  right  angles  to  the  lines  of  a 
uniform  field,  such  as  that  due  to  the  horizontal  component  of 

the  earth's  magnetic  field,  is  the  prod- 
uct of  the  strength  of  the  field  and 
the  quantity  known  as  the  magnetic 
moment  of  the  magnet.  Magnetic  mo- 
ment is  defined  as  the  product  of  the 
pole  strength  m  and  the  distance  I 
between  the  poles  of  the  magnet.  The 
application  of  the  term  "magnetic  mo- 
ment" to  this  quantity  is  appropriate, 
because  if  the  magnet  were  placed  in 
a  field  of  unit  strength,  in  which  case 
the  force  acting  on  each  pole  would 
be  m  dynes,  and  were  placed  at  right 
angles  to  the  direction  of  the  field,  as 

in  Figure  10,  the  moment  of  force  M  acting  to  turn  it  into  the 
direction  of  the  lines  of  force  would  be  given  by  the  equation 


M=ml 


(1) 


17.  Determination  of  the  strength  of  a  magnetic  field.    The 

particular  field  for  which  we  shall  now  outline  a  method  of  deter- 
mining the  strength  will  be  the  horizontal  component  of  the 
earth's  magnetic  field.  The  measurement  of  this  field  strength 
is  intimately  connected  with  the  determination  of  the  magnetic 
moment  of  a  magnet.  Neither  the  field  strength  H  nor  the  mag- 
netic moment  M  of  the  magnet  used  can  be  determined  directly 
by  a  single  experiment ;  but  two  experiments  can  be  performed, 
each  of  which  gives  a  relation  between  M  and  H.  From  these 
two  relations  both  M  and  H  can  be  found. 

One  experiment  consists  in  suspending  a  large  magnet  mm1  of 
unknown  moment  M  in  the  field  of  which  the  strength  H  is 
sought,  and  observing  the  period  of  its  oscillation.  This  gives 
an  expression  for  MH,  as  will  be  shown  in  section  20. 

The  other  experiment  consists  in  placing  the  same  magnet  mm' 
of  magnetic  moment  M  in  a  position  due  east  or  west,  as  in 


STRENGTHS  OF  MAGNETIC  FIELDS  AND  POLES      19 

Figure  11  (p.  20),  from  a  very  small  suspended  magnet  be  wrhich 
hangs  in  a  horizontal  position  in  the  earth's  field.  The  suspended 
magnet  be  is  then  acted  upon  simultaneously  by  two  fields' which 
are  at  right  angles  to  each  other :  first,  the  earth's  field  H  which 
exerts  a  moment,  or  couple,  tending  to  swing  the  needle  into  a 
north-and-south  position  ;  and  second,  the  magnetic  field  F  due 
to  the  magnet  mm'  winch  exerts  a  couple  tending  to  swing  the 
needle  into  an  east-and-west  position.  From  the  position  of  equi- 
librium assumed  by  the  small  magnet  under  the  action  of  these 
two  fields,  an  expression  for  M/H  is  Obtained,  as  will  be  shown  in 
section  19.  But  in  order  to  obtain  this  expression,  the  strength 
of  the  field  F  must  first  be  found  in  terms  of  the  magnetic 
moment  M  of  the  large  magnet. 

18.  Strength  of  field  due  to  a  magnet.  The  strength  of  the 
field  produced  at  o  (Fig.  11)  by  the  magnet  mm1  is  by  definition 
the  force  which  this  field  exerts  at  that  point  upon  unit  magnet 
pole.  If,  then,  r  is  the  distance  from  the  middle  of  the  magnet  mm' 
to  the  suspended  needle,  the  length  of  which  is  very  minute  in 
comparison  with  the  distance  r,  then  by  the  law  of  force  given  in 
section  1  the  force  which  the. pole  m  exerts  upon  unit  pole  at  the 

given  distance,  namely  (r  —  l/2),  is /79~V    Tne  otner  P°le  m' 

exerts  an  equal  and  opposite  force  of — -  dynes.    Since 

m  =  m1,  the  resultant  force,  that  is  the  field  strength  F,  is  given 
by  the  equation 


F  = 


H)'  H) 


2mlr 


2  /        l\2 

('-5) 


20  ELECTRICITY,  SOUND,  AND  LIGHT 

It  will  be  seen  from  this  equation  that  if  r  is  taken  ten  times  as 
large  as  /,  an  error  of  one  half  of  one  per  cent  is  introduced  by 
neglecting  /2/4  in  comparison  with  V.  If  r  is  twenty  times  as 
large  as  /,  this  error  is  only  one  eighth  of  one  per  cent.  Thus 
it  is  evident  that  if  r  is  taken  sufficiently  large,  no  observable 
error  is  introduced  into  F  by  writing  simply 


This  gives  the  field  strength  F  due  to  the  magnet  mmf  of  moment 
M  at  the  position  at  which  the  small  magnet  is  suspended. 

19.  Determination  of  M/H.  Returning  now  to  the  considera- 
tion of  the  deflection  which  is  produced  in  the  needle  be  (Fig.  11) 
by  the  presence  of  the  magnet  mm',  it  is  evident  that  when  the 
needle  takes  up  its  position  of  rest  it  is  in  equilibrium  under  the 


I 
I 

FIG.  11 

action  of  two  moments  of  force.  If  n  is  the  pole  strength  of 
the  suspended  magnet  be,  then,  considering  both  poles,  the  moment 
of  force  due  to  H  is  2  Hn  X  ab.  The  moment  of  force  due  to  F  is 
2  Fn  X  ao.  But  ab  =  bo  sin  0  and  ao  =  bo  cos  0.  Hence 

2  Hn  (bo)  sin  0  =  2  j^w  (&o)  cos  0,  (3) 

or  Htsmct)=F'  (4) 


M  /c 

or  from  (2)                                 _  =  ___.  (5) 

This  gives  the  first  relation  between  M  and  H  in  terms  of   the 
measurable  quantities  </>  and  r. 


STKENGTHS  OF  MAGNETIC  FIELDS  AND  POLES      21 


20.  Determination  of  ME.  The  second  relation  between  M  and 
H  is  obtained  from  an  observation  of  the  period  of  vibration  of 
the  magnet  mm'  when  it  is  suspended  at  the  point  for  which  H 
is  to  be  determined,  namely  the  position  occupied  above  by  the 
small  magnet  be.  In  order  to  find  this  period  it  is  first  necessary 
to  find  what  relation  exists  between  the  moment  of  force  Fh  which 
acts  to  restore  the  magnet  to  its  north-and-south  position  at  any 
instant  at  which,  in  the  course  of  its  vibration,  it  is  displaced  from 
this  position  by  an  angle  6.  It  is  evident  at  once  from  Figure  12 
that  this  restoring  moment  is  given  by 

~Fh  =  1  Hm  X  de  =  Hm  X  df  =  Hml  sin  6  =  MH  sin  6.        (6) 

This  equation  shows  that  in  general  the  restoring  moment  is  not 
proportional  to  the  angle  of  displacement  6,  but  rather  to  sin  6. 
If,  however,  we  keep  the  angle  of  swing  very  small,  sin  6  will  not 
differ  sensibly  from  0,  and  we  may  .write  Hm 

Wi=MH0,      or      ^  =  MH.          (7) 


Now  a  simple  harmonic  vibration  is  defined 
as  one  in  which  the  restoring  moment  of  force 
acting  upon  the  vibrating  system  is  always  pro- 
portional to  the  angle  of  displacement  of  the 
system  from  its  position  of  rest.  We  see,  there- 
fore, that  so  long  as  the  amplitude  of  vibration 
of  the  magnet  is  small,  its  motion  is  a  case  of 
simple  harmonic  motion ;  and  we  may  therefore 
apply  to  it  the  general  formula  for  the  period 
of  any  simple  harmonic  motion.  If  t  represents 
the  period  of  a  half  vibration,  /the  moment  of 
inertia  of  the  vibrating  system,  and  Fh/6  the 
force  constant  of  the  system  (i.e.  the  constant  ratio  of  the  moment  of 
force  and  the  displacement),  then  the  general  equation  for  any  simple 
harmonic  motion  of  rotation  is 


- 


*  See  "  Mechanics,  Molecular  Physics,  and  Heat,"  pp.  87-91  ;  also  pp.  74  and  75. 


22  ELECTRICITY,  SOUND,  AND  LIGHT 

Since  by  equation  (7)  the  force  constant  of  the  vibration  under 
consideration  is  MH,  we  have  at  once  as  the  expression  for  the 
period  ^  of  a  half  swing  of  the  magnet 


or         MH=.  (9) 

Cl 

This  equation  gives  us  the  second  relation  between  M  and  If  in 
terms  of  the  measurable  quantities  tl  and  /. 

21.  Expression  for  H.  From  the  two  equations  (5)  and  (9) 
either  M  or  H  can  easily  be  obtained.  Thus  the  division  of 
(9)  by  (5)  gives  2 


22.  Moment  of  inertia  of  the  magnet.  In  order  to  determine 
experimentally  /,  the  moment  of  inertia  of  the  large  magnet 
(mm'  of  Fig.  11),  it  is  most  -convenient  to  lay  upon  it  a  brass 
ring  of  known  mass  W  and  known  radius  R  (Fig.  14,  p.  26)  ;  to 
adjust  it  until  its  center  coincides  with  the  axis  of  suspension  ; 
and  then  to  take  the  period  t2  (for  small  vibrations)  of  the  new 
system.  The  moment  of  inertia  of  this  new  system  is  now 
/+  J0,  J0  being  the  known  moment  of  inertia  of  the  ring.  Hence 
the  formula  for  the  new  period  is 


The  solution  of  (9)  and  (11)  for  /gives 


The  moment  of  inertia  of  the  ring  is  simply  WR2,  where  W  is  the 
mass  in  grams,  and  R  the  mean  radius  in  centimeters  of  the  ring. 
For  it  is  obvious  that  the  general  formula  for  moment  of  inertia, 
namely  2wr2,  in  which  w  represents  an  element  of  mass  and  r  its 
distance  from  the  axis,  reduces  in  the  case  of  a  ring,  for  which  all 
the  elements  are  at  the  same  distance  from  the  center,  to  WRZ.  * 


2\ 

*  The  rigorous  formula  for  the  moment  of  inertia  of  a  ring  is  WA--s  —  ~) 
in  which  RI  and  R%  are  the  inner  and  outer  diameters. 


STRENGTHS  OF  MAGNETIC  FIELDS  AND  POLES      23 

As  a  check  upon  the  experimental  determination  of  /  it  is  well 

W'il2  +  52)   * 
to  take  also  the  theoretical  value,  namely  —  ^—  —  —  ->     where  W 

\.Zi 

is  the  mass  of  the  magnet,  I  its  length,  and  b  its  width,  f 

23.  Comparison  of  magnetic  fields.  From  equation  (9)  it  is 
evident  that  if  ^  and  t2  represent  the  half  periods  of  the  same 

*  See  "Mechanics,  Molecular  Physics,  and  Heat,"  pp.  78-81. 

t  Correction  of  the  formulas  for  torsion  of  the  suspending  fibers.  Equation  (3) 
was  deduced  upon  the  assumption  that  no  forces  other  than  magnetic  ones  were 
concerned  in  the  equilibrium  of  the  magnet  when  its  deflection  was  represented 
by  the  angle  0  (Fig.  11).  But  evidently  the  torsion  of  the  suspending  fiber,  as 
well  as  the  field  Zf,  opposes  the  deflection  due  to  the  field  F.  This  torsional 
element  cannot  always  be  neglected.  In  order  to  take  it  into  account  we  have 
only  to  consider  that  by  Hooke's  law  there  is  a  constant  ratio  between  the 
restoring  torsional  moment  Fh'  and  the  angle  of  displacement.  It  is  customary 
to  define  this  ratio  as  the  moment  of  torsion  of  the  suspension  and  to  denote  it  by 
TQ.  Thus,  by  definition, 

—  =r0,         or        Fh7=TQ<t>.  »       (13) 

0 

Thus  when  the  magnetic  needle  be  (Fig.  11)  was  deflected  through  an  angle  0 
the  restoring  torsional  couple  was  To0.  Hence  the  rigorous  equation  of  equi- 
librium was  not  (3),  but  rather 

2  Hn  (bo)  sin  0  +  T0(f>  =  2Fn  (bo)  cos  0.  (14) 

Now  TO  can  be  obtained  by  a  second  experiment  as  follows:  The  pin  to 
which  the  upper  end  of  the  supporting  fiber  is  attached  is  always  arranged 
in  magnetometers  so  that  it  can  be  rotated  through  any  desired  angle  about  a 
vertical  axis.  If,  then,  this  so-called  torsion  head  (see  t,  Fig.  13),  be  rotated 
through,  say,  180°  (TT  radians),  the  needle,  which  was  before  in  the  magnetic 
meridian,  will  now  be  deflected  by  the  torsion  of  the  fiber  alone  through  an 
angle  of,  say,  a  radians.  Since  the  actual  twist  in  the  fiber  is  then  (TT  —  a] 
radians,  the  equation  of  equilibrium  under  these  conditions  is  evidently 

2  ffn  (60)  sin  «  =7-,,  (*-«),        or        T,  = 


Substituting  this  value  for  T0  in  (14)  and  dropping  the  common  terms,  we  get 
IT  sin  0  +  Hsinal  —  -  —  )  =  Fcos0, 

•          \  (1 

ir  sin,  i  +  Ei.  _-)=*•  cos*. 


Now  since  the  experiment  should  always  be  arranged  so  that  both  a  and  0 
are  small  (if  a  is  more  than  5°  the  torsion  head  may  be  twisted  through  90° 


24  ELECTRICITY,  SOUND,  AND  LIGHT 

magnet  in  two  different  fields  of  horizontal  intensities  H^  and  Hz 
respectively,  then  u       ^ 


EXPERIMENT  2 

(A)  Object.    To  determine  the  ratio  M/H. 

Directions.  The  apparatus  used,  commonly  known  as  a  magnetometer, 
is  shown  in  Figure  13. 

I.  Set  the  suspended  magnetic  needle  nn'  in  the  magnetic  meridian. 
To  do  this  set  up  a  reading  telescope  and  scale  in  front  of  the  small 
mirror  m  and  two  or  three  meters  distant  from  it.  The  mirror  and 


instead  of  180°),  no  appreciable  error  is  introduced  by  replacing  sin  a-/  sin  0 
by  a/0.     Equation  (1C)  then  reduces  to 


F.  (17) 

Substitution  in  equation  (5)  gives 

M  _  r*  tan  0        TT 

ff~  T"i^' 

The  final  equation  for  II,  namely  (10),  then  becomes 

.  (10) 


Rigorously,  t  in  equations  (10)  and  (19)  also  needs  correction  for  the  torsion 
of  the  fiber  which  supports  the  magnet  when  its  period  is  determined.  But  since 
the  magnet  mm'  is  large,  if  the  fiber  is  properly  chosen  this  correction  is  unnec- 
essary. In  case  it  should  be  needed  the  method  by  which  it  is  obtained  and 
applied  may  be  seen  from  the  following. 

The  correct  equation  for  the  half  period  is 

or      t*  (Ma  +T$  =  •**!,  (20) 

where  TO  is  the  moment  of  torsion  of  the  thread  which  holds  the  large  magnet. 
If  twisting  the  torsion  head  of  this  magnet  support  through  an  angle  of  TT  radians 
causes  the  magnet  to  turn  through  [3  radians,  we  have,  since  /3  is  very  small, 
TO(TT  —  /3)  =  MHp.  Substitution  of  this  value  of  TO  in  the  above  equation  (20) 


The  combination  of  this  with  the  equation  (18)  found  above  gives  as  the 
final  corrected  form  for  the  value  of  H 

ft 


STRENGTHS  OF  MAGNETIC  FIELDS  AND  POLES      25 

magnet  are  rigidly  connected,  the  latter  usually  consisting  of  a  bit  o^ 
magnetized  needle  a  few  millimeters  long  attached  by  shellac  directly 
to  the  back  of  the  mirror.  Focus*  the  telescope  upon  the  reflected 
image  of  the  scale.  Then  so  adjust  the  torsion  head  t  that  equal  deflec- 
tions of  the  head  to  the  right  and  to  the  left  produce  equal  deflections 
upon  the  scale. 

II.  Place  the  magnet  mm'  upon  the  graduated  crossbar  of  the  magnet- 
ometer at  such  a  distance  east  of  the  suspended  magnet  nn'  that  r/l  =  10 
(see  p.  20)  ;  then  read  with  the  telescope  and  scale  the  deflection  produced 
by  turning  mm'  end  for  end.  Repeat  this  operation  several  times,  and  thus 
determine  the  order  of  accuracy  of  the  observation. 


C 


FIG.  13 

III.  Make  the  same  observation  when  mm'  is  at  an  equal  distance  to 
the  west  of  the  suspended  magnet.  From  the  mean  of  these  two  deflec- 
tions determine  tan  <£.  This  is  found  with  sufficient  accuracy  by  dividing 
the  deflection,  measured  in  centimeters,  by  four  times  the  distance  L  from 
the  scale  to  the  mirror  of  the  suspended  magnet.  The  reason  for  this  will 
be  clear  when  it  is  remembered  that  the  beam  of  light  turns  through  twice 
as  large  an  angle  as  the  mirror,  and  that  turning  the  magnet  mm'  end  for 
end  produces  a  deflection  of  2  0  in  the  mirror. 


*  See  directions  for  focusing  the  telescope,  as  given  on  page  64. 


26 


ELECTRICITY,  SOUND,  AND  LIGHT 


IV.  Remove  mm"  and  determine   a  (if  necessary)   as  outlined  in  the 
theory  of  footnote  f,  page  23.    The  angle  a  is  found  in  radians  by  divid- 
ing the  scale  deflection  by  2  L.    Take  the  mean  of  the  two  results  obtained 
by  twisting  the  torsion  head  equal  distances  in  opposite  directions. 

V.  Obtain  a  value   for  tan<£  for  another   position  of   mm'  such  that 
r/l  =  14.    Calculate  the  value  of  M/H  for  each  deflection. 

(B)  Object.   To  determine  the  value  of  MH  with  the  same  magnet  that  was 
used  in  (A),  and  for  the  same  locality,  and  by  combination  of  the  results 

with  those  of  (A)  to  find  H. 

Directions.  I.  Eliminate  tor- 
sion from  the  suspending  fiber 
of  mm'  by  turning  the  torsion 
head  (Fig.  14)  until  the  stir- 
rup comes  to  rest  approxi- 
mately in  the  meridian.  Then 
suspend  mm'  from  the  fiber 
by  placing  it  in  this  stirrup 
so  that  it  rests  accurately 
horizontal.  Take  the  pefiod 
of  the  magnet  with  the  stop 
watch,  using  a  telescope  and 
scale.  Make  three  observa- 
tions of  the  time  of  thirty 
half  oscillations.  The  ampli- 
tude should  not  exceed  3°, 
and  the  instants  of  passage 
through  the  mid-points,  not 
through  the  end  points,  of  the 
swing  should  be  observed. 

II.  From  the  dimensions  of 
the  magnet  mm',  and  from  its 
mass    as    found    by    a    rough 
balance,  determine  /.     If  time 
permit  check  this  by  the  experi- 
mental method  of  section  22. 

III.  Calculate  H  from  each 
of  the  mean  values  of  tan  0  and  the  corresponding  value  of  r. 

(C)  Object.    To  find  the  value  of  H  in  a  different  locality  by  comparison 
with  the  value  found  in  (B). 

Directions.  Set  up  the  magnet  and  suspension  of  (B)  at  a  point  to  be 
indicated  by  the  instructor  (the  proposed  location  of  the  tangent  galvanom- 
eter of  Experiment  3).  Make  three  observations  of  the  time  of  thirty  half 
oscillations.  From  equation  (23)  and  the  value  of  Hl  and  tl  obtained  in  (B) 
calculate  7/2  for  this  position. 


FIG.  14 


STRENGTHS  OF  MAGNETIC  FIELDS  AND  POLES      27 

EXAMPLE 

(A)  The  distance  in  room   No.  9   of   mm'  west  of    the   small   magnet 
mounted  on  the  mirror  of  the  magnetometer  was  100  cm.     The  distance 
of    the    telescope    and    scale    from    the    mirror  was    4.13   cm.     The    first 
reversal  of  the  magnet   changed    the    scale    reading   from   20.25  cm.   to 
42.55  cm.,  the   second   from   42.70  to  20.40,  and  the  third  from  20.40 
to  42.50,  thus  making  the  mean  deflection  at  100  cm.,  22.20  cm.,  and  the 
mean  value  of  tan0  22.20  -H  (4  x  413).    Similarly  the  mean  value  of  tan0 
at  a  distance  of  140  cm.  was  8.12  -r-  (4  x  413).    Twisting  the  torsion  head 
through  180°  produced  a  change  in  the  scale  reading  of  3.1  cm.,  thus 
making  the  value  of  a  3.1  -f-  (2  x  413)  radians. 

(B)  Three  observations  on  the  time  of  thirty  half  swings  of  mm'  when 
placed  in  the  position  occupied  by  nnf  above  gave  148.4  sec.,  148.6  sec., 
and  148.4  sec.     Hence  tl  =  4.947.    The  mass  of  the  magnet  was  61.29  g., 
its  length  10  cm.,  and  its  breadth  1.3  cm.    The  value  of  7  computed  from 
these  dimensions  was  518.6.     When  a  40  g.  ring  of  4  cm.  mean  radius 
was  added  the  half  period  was  7.40  sec.    The  value  of  /  computed  from 
these    observations   was    517.8.     H  computed   from   the  observations   at 
r=100  was  .1764;  from  the  observations  at  r  =  140,  .1757,  thus  giving 
a  mean  value  in  room  No.  9  of  .1760.    The  value  of  M  was  1185,  and  the 
number  of  units  of  magnetism  in  each  pole  was  150,  approximately. 

(C)  The  half  period  of  the  magnet  in  room  No.  19  was  4.818  sec.    Hence 
the  mean  value  of  H  in  room  No.  19  was  .1856  gausses. 


CHAPTER  III 
MEASUREMENT   OF  ELECTRIC  CURRENTS 

24.  Definition  of  the  electric  current.  The  charges  (or  quan- 
tities) of  electricity  described  and  denned  on  page  2,  section  2, 
by  means  of  certain  attractive  and  repellent  properties  which 
they  possess,  are  found  to  exhibit  new  properties  as  soon  as  they 
are  set  into  motion.  An  electric  charge  in  motion  is  called  an 
electric  current. 

There  are,  however,  two  ways  in  which  the  charge  may  move. 

Suppose,  for  instance,  that  the  given  charge  is  contained  upon  the 

small  round  body  A  (Fig.  15),  and  that  the  body  with  its  charge 

b    is  rapidly  carried  to  B,  drawn  perhaps 

by  an  opposite  charge  upon  the  rod  ab. 

Then,  according  to  the. above  definition, 

an  electric  current  has  passed  from  A  to  B. 
FlG  15  ""  Very   good   reasons   exist  (see  sect.  31, 

p.  38)  for  supposing  that  electric  currents 

in  liquids  (electrolytes)  are  of  this  nature.  The  current  probably 
consists  of  a  swarm  of  charged  particles  of  matter  (ions)  actually 
moving  through  the  liquid  under  the  influence  of  an  electric  field, 
just  as  the  charged  pith  ball  moves  through  the  air. 

But  we  have  seen  in  section  3  that  there  is  another  way  in 
which  the  charge  upon  A  may  arrive  at  B.  Thus  when  an  un- 
charged body  is  placed  at  B  and  then  connected  by  a  wire  with 
the  charged  body  A  (Fig.  16),  the  instant 

the  connection  is   made   B  is   found   to    (~~\ /"""^ 

possess  the  same  sort  of  attractive  prop-  pIG.  10 

erties  which  were  before  possessed  by  A, 

while  A  is  found  to  have  lost  some  of  its  attractive  power.  The 
electric  charge  upon  A  (or  part  of  it)  has  moved  along  the  wire 
to  B.  According  to  the  electron  theory,  in  this  case,  also,  a 

28 


MEASUREMENT  OF  ELECTRIC   CURRENTS          29 

swarm  of  very  minute  charged  particles  has  actually  streamed 
through  the  metal  from  A  toward  B,  or  vice  versa;  but  since 
no  direct  proof  of  the  correctness  of  this  view  has  yet  been 
found,  it  must  be  regarded  at  present  as  merely  an  hypothesis. 
Nevertheless  it  is  certain  that  a  part  of  the  charge  which  was 
at  first  upon  A  has  actually  passed  in  some  way  along  the  wire 
to  B.  Whether,  then,  a  charge  passes  from  A  to  B  by  the  first 
method,  in  which  matter  is  observed  to  move  with  the  charge, 
or  by  the  second  method,  in  which  minute  particles  are  only 
assumed  as  the  carriers  of  the  charge,  it  is  the  fact  of  its 
passage  between  A  and  B  which  ive  shall  consider  as  consti- 
tuting an  electric  current. 

That  the  properties  possessed  by  a  static  charge  which  is 
made  to  move  rapidly  through  space  by  mechanical  means  are 
in  fact  identical  in  all  respects  with  the  properties  possessed 
by  a  charge  which  is  moving  along  a  wire  (see  sect.  25)  was 
first  proved  in  an  elaborate  investigation  made  in  1876  by 
the  American  physicist  Henry  A.  Rowland.  The  correctness 
of  this  conclusion  has  been  confirmed  by  many  subsequent 
investigations. 

25.  Magnetic  effect  of  an  electric  current.  The  new  property 
possessed  by  a  charge  in  motion,  but  not  possessed  by  a  charge 
at  rest,  is  the  property  of  exerting  magnetic  influences.  That  a 
charge  at  rest  does  not  produce  any  magnetic  effect  may  be 
clearly  demonstrated  as  follows.  If  a  charged  body  is  brought 
east  or  west  of,  and  near  to,  a  magnet  supported  upon  a  point, 
the  needle  will  indeed  at  first  swing  about  toward  the  charged 
body  as  though  the  latter  exerted  a  magnetic  effect  upon  it. 
That  this  is  in  reality,  however,  an  effect  due  merely  to  elec- 
trostatic induction  may  be  convincingly  shown  by  inserting  a 
sheet  of  copper,  zinc,  or  aluminum  between  the  magnet  and 
the  charge,  when  the  former  will  be  found  to  swing  back  at 
once  to  its  north-and-south  position.  For  the  sheet  cuts  off 
all  electrostatic  influences  in  accordance  with  the  principle  of 
electric  screening  (sect.  13) ;  but  it  has  no  influence  at  all  upon 
magnetic  forces,  as  may  be  shown  by  inserting  it  between  a 
magnet  and  the  suspended  needle. 


30  ELECTRICITY,  SOUND,  AND  LIGHT 

Now  if  a  strong  static  charge,  for  example  that  upon  a  Leyden 
jar,  is  passed  through  a  wire  which  is  wound  one  or  two  hun- 
dred times  around  a  small  glass  tube  containing  an  unmagiietized 
knitting  needle,  the  needle  will  be  found  to  have  been  quite 
strongly  magnetized.  If  the  sign  of  the  charge  is  reversed  and 
the  knob  of  the  jar  touched  to  the  same  end  of  the  wire  as  at 
first,  a  second  needle  will  in  general  be  found  to  have  been  oppo- 
sitely magnetized.  This  certainly  shows  that  a  charge  in  motion 
produces  some  sort  of  a  magnetic  effect. 

Again,  if  the  positive  terminal  of  a  static  machine  is  connected 
to  one  terminal  of  a  coil  of  wire  of '  many  turns  in  the  middle  of 
which  hangs  a  magnetic  needle,  the  needle  will  be  deflected  when 
the  machine  is  in  operation ;  and  if  the  terminals  of  the  machine 
are  reversed,  the  direction  of  deflection  of  the  needle  will  be 

reversed  also. 

sr\  s^\         A  more  convenient  way,  however,  of 

showing  that  a  charge  in  motion  pro- 
duces a  magnetic  effect  is  to  make  use 
of  a  galvanic  cell. 

Such  a  cell  consists  essentially  of  two 
2n  dissimilar   conducting   solids    immersed 

in  any  conducting  liquid  (the  liquid  can- 
not be  a  molten  metal).  It  may  be 
looked  upon  as  a  self-acting  static  machine.  For  if  the  bodies  A 
and  B  be  connected  to  the  two  plates  of  such  a  cell  (see  Fig.  17), 
they  are  found  to  be  statically  charged,  just  like  the  poles  of  a 
static  machine,  the  one  connected  to  the  copper  or  carbon  being 
positive,  and  the  one  connected  to  the  zinc,  negative.  (It  requires, 
however,  a  delicate  electroscope  to  prove  the  existence  of  these 
static  charges  upon  the  terminals  of  a  galvanic  cell.)  If  now  A 
and  B  are  connected  by  a  wire,  the  positive  charge  upon  A  dis- 
charges to  B  (or  if  it  is  preferred  so  to  consider  it,  B  loses  its 
negative  charge  to  A)-  but  the  chemical  action  which  is  set  up 
in  the  cell  recharges  A  (or  B)  as  fast  as  it  is  discharged.  Hence 
the  galvanic  cell,  like  the  continuously  turned  static  machine, 
produces  in  the  wire  a  continuous  current  when  its  terminals  are 
connected  by  a  conductor. 


Cu 
FIG.  17 


MEASUKEMENT  OF  ELECTRIC  CURRENTS         31 

If  the  wire  connecting  the  terminals  of  such  a  cell  is  held  over 
and  parallel  to  a  magnetized  needle,  the  latter  will  be  strongly 
deflected  (Fig.  18).  If  the  terminals  of  the  cell  are  interchanged 
the  deflection  will  be  reversed.  This  experiment  was  first  per- 
formed by  the  Danish  physicist  Oersted  in  1819.  It  constitutes 
one  of  the  most  important  discoveries  which  has  ever  been  made 
in  the  history  of  science,  for  it  established  for  the  first  time  a 
connection  of  some  sort  between  electricity  and  magnetism  and 
paved  the  way  for  the  marvelous  electrical  developments  of  the 
nineteenth  century. 

The  preceding  experiments  have  shown  that  reversing  the 
direction  in  which  the  charge  (or  charges)  moves  past  the  needle 
reverses  the  magnetic  effect  observable  near  the  path  of  the  charge. 
According  to  the  two-fluid  theory 
the  current  consists  in  the  mo- 
tion of  a  positive  charge  in  one  H 
direction  and  a  negative  charge 
in  the  other  direction.  According 
to  Franklin's  one-fluid  theory  it 
consists  of  a  motion  only  of  posi-  FlG  18 

tive  electricity  in  one  direction. 

According  to  the  electron  theory  it  consists  of  a  motion  only  of 
negative  electrons  in  the  opposite  direction.  It  is  wholly  immaterial 
which  of  these  theories  is  correct  so  long  as  we  understand  the 
conventions  which  are  in  common  use  regarding  direction  of  cur- 
rent. According  to  universal  convention  the  direction  of  a  current 
is  the  direction  from  positive  toward  negative,  i.e.  in  the  case  of  a 
galvanic  cell  from  copper,  or  carbon  (in  the  external  circuit), 
toward  zinc.  Thus,  according  to  this  convention,  when  B  was 
negatively  charged  and  A  uncharged,  joining  them  with  a  wire 
caused  a  momentary  electric  current  to  flow  from  A  to  B,  not 
from  B  to  A.  This  definition  makes  it  unnecessary  to  introduce 
the  terms  "positive  current"  and  "negative  current,"  although 
these  terms  are  sometimes  employed. 

26.  Form  of  magnetic  field  about  a  conductor.  The  form  and 
direction  of  the  magnetic  field  which  surrounds  a  current  may  be 
obtained  by  mapping  out  the  field  with  a  compass  needle  in  the 


32 


ELECTRICITY,  SOUJSD,  AND  LIGHT 


FIG.  19 


manner  described  in  Experiment  1,  (A).    Thus  if  the  black  center 

of  Figure  19  represents  the  cross  section  of  a  conductor  which 

passes  vertically  through  the  paper  and  through  which  the  cur- 

rent flows  from  the  reader  toward 
the  plane  of  the  paper,  as  indicated 
by  the  cross  in  the  middle  of  the 
conductor,*  then  the  map  of  the  mag- 
netic field  in  the  plane  of  the  paper 
is  that  shown  in  the  figure.  The 
lines  of  force  are  circles  about  the 
current  as  a  center,  and  the  direc- 
tion of  these  circles,  i.e.  the  direc- 
tion in  which  an  isolated  N  pole 
would  rotate  about  the  current,  is 
from  left  to  right,  as  the  observer 
looks  in  the  direction  of  the  current. 

This  rule  is  often  known  as  the  right-handed-screw  rule,  for  the 

relation  between  the  direction  of   rotation  of  the  lines  and  the 

direction  of  motion  of  the  current 

is  the  same  as  that  existing  between 

the  rotary  and  forward  motions  in 

a  right-handed  screw.    The  analogy 

stops  here,  however,  for  the  lines  of 

force  are  not  spirals  about  the  current. 

The  plane  of  their  direction  is  always 

perpendicular  to  that  of  the  current. 
It  is  to  be  expected  from  the  above 

rule  that  an  isolated  pole  would  con- 

tinue to  move  indefinitely  in  a  circu- 

lar path  around  a  conductor  carrying 

a  current.    That  this  is  indeed  the 

case  may  be  strikingly  shown  by  the 

following  experiment.    A  vertical 


X 


FlG  2o 


*  In  general  a  dot  in  the  middle  of  the  cross  section  of  a  conductor  is  used 
to  represent  the  head  of  an  approaching  arrow,  i.e.  a  current  flowing  toward 
the  reader.  A  cross  in  the  conductor  represents  the  tail  of  a  retreating  arrow, 
i.e.  a  current  flowing  away  from  the  reader. 


MEASUREMENT   OF  ELECTRIC   CURRENTS          33 

conductor  ac  (Fig.  20),  free  to  turn  about  its  own  axis,  carries  a  bar 
magnet  NS.  A  circular  mercury  cup  bb',  shown  in  cross  section, 
admits  of  a  continuous  connection  (over  the  path  be)  of  the  ter- 
minals a  and  c  with  a  battery  B.  A  current  is  now  allowed  to 
flow  from  a  to  c.  This  current  produces  no  field  at  the  point  S 
and  therefore  may  be  considered  as  acting  011  an  isolated  pole  N. 
Under  its  action  the  pole  N  will  actually  be  found  to  rotate 
indefinitely  in  the  direction  determined  by  the  right-handed-screw 
rule.  If  the  direction  of  the  current  is  reversed,  the  direction  of 
the  rotation  is  found  to  be  reversed  also.  As  is  evident  from  the 
figure,  connection  with  the  battery  may  be  made  at  b  and  d  and 
the  experiment  performed  with  the  S  instead  of  the  N  pole. 

27.  The  unit  of  current  strength.  Since  an  electric  current  is 
defined  as  an  electric  charge  in  motion,  it  is  natural  to  measure 
the  strength,  or  intensity,  of  current  flowing  through  a  conductor 
by  the  number  of  units  of  charge  which  pass  through  any  cross 
section  of  the  conductor  per  second.  This  is  indeed  the  definition 
of  current  strength  in  the  so-called  electrostatic  system.  Thus  if  Q 
represents  the  number  of  units  of  charge  (electrostatically  measured, 
sect.  2,  p.  2)  which  pass  through  the  conductor  in  t  seconds,  then 
the  mean  current  intensity  /  is  given  by  the  equation 

z-f-  (D 

The  current  flowing  at  any  instant  is  the  rate  of  passage  of  quan- 
tity, i.e.  if  we  let  dQ  represent  the  quantity  of  electricity  which 
passes  a  given  cross  section  in  the  short  element  of  time  dt,  then 
the  rigorous  definition  of  current  is  given  by 

'-%• 

A  wire  then  carries  a  unit  of  current  when  a  charge  passes  through 
it  at  the  rate  of  unit  quantity  per  second. 

But  Oersted  was  experimenting  with  a  galvanic  cell  when  he 
discovered  the  magnetic  effect  of  a  current,  and  neither  he  nor 
his  contemporaries  realized  that  a  current  was  simply  a  charge  in 
motion,  and  that  the  magnetic  effect  could  be  produced  just  as 
well  by  means  of  the  electricity  developed  by  means  of  a  static 


34  ELECTRICITY,  SOUND,  AND  LIGHT 

machine.  Hence  another  unit  of  current  strength  was  chosen,  and 
one  which  had  no  apparent  connection  with  the .  unit  of  electro- 
static quantity  defined  on  page  3.  The  size  of  the  magnetic 
effect  was  arbitrarily  taken  as  the  measure  of  current  strength. 
But  this  effect  was  found  to  vary  both  with  the  length  of  the 
conductor  and  with  the  distance  from  the  conductor  to  the  point 
at  which  it  was  measured.  Hence  it  was  necessary  to  fix  upon  a 
wire  of  specified  length  and  to  measure  the  magnetic  effect  at  a 
specified  distance  from  it.  A  wire  1  cm.  long  was  chosen,  and  in 
order  to  arrange  to  have  all  parts  of  this  wire  at  the  same  distance 
from  some  point  at  which  the  magnetic  effect  was  to  be  measured, 
it  was  decided  to  bend  it  into  an  arc  of  1  cm.  radius  and  to  meas- 
ure the  magnetic  effect  at  the  center  of  this  arc.  Thus  a  current 

of  unit  strength  was  said  to  be  flowing 
icm  J^      in  the  wire  when  a  length  of  this  wire 

equal  to  1  cm.  and  bent  into  an  arc  of 
1  cm.  radius  created  at  the  center  of  this 
s        /  arc  a  magnetic  field  of   unit   strength. 

\     £     /  Otherwise   stated,  unit  current  is   that 

/  current    unit    length    of    which    placed 

everywhere  at  unit  distance  from  unit 
magnetic  pole  acts  upon  it  with  a  force 
of  1  dyne.  Thus  if  the  arc  AB  is  1  cm. 

and  if  r  is  1  cm.  (see  Fig.  21),  then  /  is  one  unit  when  the 
magnetic  field  strength  at  0  is  1  dyne.  Similarly  /  is  10  when 
the  field  strength  at  0  is  10  dynes,  etc. 

Thus,  in  this  so-called  electro-magnetic  system,  it  is  the  unit  of 
current  strength  which  is  first  defined  and  which  is  therefore  the 
fundamental  unit.  The  unit  of  quantity  in  this  system  is  defined 
as  the  quantity  conveyed  per  second  through  every  cross  section 
of  a  conductor  which  carries  a  current  of  unit  strength.  Thus,  just 
as  7  —  Q/t  was  the  equation  which  defined  current  in  the  electro- 
static system,  Q  having  been  first  defined,  so  the  same  equation 
written  in  the  form  It  —  Q  defines  quantity  in  the  electro-magnetic 
system,  /  having  been  first  defined.  Thus  100  units  of  quantity 
pass  in  10  seconds  through  a  conductor  which  carries  a  constant 
current  whose  strength  is  10  units. 


MEASUREMENT   OF  ELECTKIC   CURRENTS          35 

28.  Ratio  of  the  electrostatic  and  the  electro-magnetic  units  of 
quantity,  or  of  current.    It  is  of  interest  to  inquire  which  of  these 
two  units,  the  electrostatic  or  the  electro-magnetic,  is  the  larger. 
In  order  to  answer  this  question  experimentally,  it  is  evident  that 
it  would   only  be   necessary  to  collect  a  quantity  of   electricity 
which  had  been  measured  in  electrostatic  units  by  an  observation 
of  its  attraction  upon  some  known  charge,  and  then  to  discharge 
this  quantity  in  a  known  time  through  the  arc  of  Figure  21,  and 
measure  the  magnetic  effect  thus  produced.    Such  a  measurement 
shows  that  the  electro-magnetic  unit  is  enormously  larger  than  the 
electrostatic  unit,  and  further  reveals  the  surprising  fact  that  the 
ratio  of  the  electro-magnetic  to  the  electrostatic  unit  is  equal  to 
the  velocity  of  light  expressed  in  centimeters  per  second,  namely 
3  X  1010.    This  fact  played  an  important  role  in  the  discovery  of 
the  intimate  relation  which  exists  between  electricity  and  light. 

29.  Practical  units.    No  name  lias  been  given  to  the  absolute 
units  of    current  and  quantity    defined  above.     For  purposes  of 
convenience  it  has  been  decided  to  take  as  the  commercial  units 
one  tenth  of  the  absolute  units  of  both  current  and  quantity.    Thus 
the  commercial  unit  of  current  is  10"1  absolute  electro-magnetic 
units  of  current.     It  is  named  an  ampere  in  honor  of  the  French 
physicist  Andre  Marie  Ampere  (1775-1836).    Similarly  the  com- 
mercial unit  of  quantity  is  10"1  absolute  electro-magnetic  units  of 
quantity.    It  is  named  a  coulomb  hi 

honor  of  the  French  physicist  Charles 
Augustin  Coulomb  (1736-1806). 

30.  Tangent  galvanometer.  It  is 
evident  from  the  definition  of  current 
strength  that  the  direct  method  of 
measuring  /  must  consist  in  meas- 
uring the  strength  of  magnetic  field 
produced  by  a  known  length  of  the 
current  at  a  known  distance  from  it. 
This  is  most  easily  accomplished 
by  passing  the  current  through  a 

circular  loop  of  wire  of  known  radius  and  measuring  the  field 
strength  at  the  center  of  the  loop.    If  the  current  were  of  unit 


36  ELECTKICITY,  SOUND,  AND  LIGHT 

strength,  then  from  the  definition  of  current  strength  it  would 
follow  that  every  centimeter  of  length  of  the  conductor  carrying 
the  current  would  produce  at  a  point  which  is  one  centimeter 
distant  from  all  parts  of  that  centimeter  of  length  a  field  strength 
of  one  dyne.  Since  all  forces  which  emanate  from  point  sources 
vary  inversely  as  the  squares  of  the  distances  from  the  sources,  it 
is  evident  that  the  forces  emanating  from  points  on  the  wire 
vary  inversely  as  the  squares  of  the  distances  from  those  points. 
Hence  at  the  center  of  the  loop,  a  point  which  is  r  centimeters 
distant  from  all  points  on  the  wire,  the  field  strength  due  to  each 
centimeter  of  length  must  be  l/r2  and  the  field  strength  due  to 

2  irr 
the  2  TTT  centimeters  of  length  must  therefore  be  -  —  •     If  the 

r 
current  has  J  units  of  strength,  and  if  there  are  N  loops  instead 

of  one,  the  field  strength  F  at  the  center  must  be,  therefore, 

N      .  Hn  2  7TNI 

F  =  —       -  gausses.  (3) 

In  order  to  measure  this  field  the  plane 
of  the  coil  (see  Fig.  22)  is  set  in  the  earth's 
magnetic  meridian  and  a  small  suspended 
magnetic  needle  is  placed  at  the  center. 
In  this  position  the  field  F  due  to  the 
current  tends  to  turn  the  needle  into 
the  east-and-west  direction,  while  the 
earth's  magnetic  field  tends  to  keep  it 

in  the  north-and-south  direction.    The  equation  of  equilibrium  is, 

therefore  (see  Fig.  23  and  sect.  19), 

Hnl  sin  6  =  Fnl  cos  0,  (4) 

in  which  I  is  the  length  of  the  small  magnet. 
Hence  from  (3) 

or       /=J^Ltanft  (5) 


Thus  if  the  galvanometer  is  set  up  at  some  point  for  which  H  is 
known,  the  current  strength  /  is  determined  in  absolute  units 
from  a  measurement  of  r,  N,  and  tan#. 


MEASUREMENT   OF  ELECTRIC   CURRENTS          37 

31.  Electrolysis.  It  is  found  that  all  of  the  liquids  (save 
molten  metals)  which  are  conductors  of  electricity  are  solutions 
of  chemical  compounds,  —  salts,  acids,  or  bases,  —  and  that  the 
passage  of  an  electric  current  through  such  liquids  is  uniformly 
accompanied  by  the  passage  out  of  the  solution  of  the  compo- 
nents of  the  dissolved  substance.  Thus  when  a  current  is  passed 
through  a  solution  of  copper  sulphate  (CuS04),  metallic  copper 
(Cu)  is  deposited  upon  the  plate  by  which  the  current  leaves  the 
solution  (the  negative  electrode  or  cathode},  while  S04  radicals 
collect  about  the  positive  electrode  or  anode,  where  their  pres- 
ence may  be  detected  by  the  acid  character  which  they  impart  to 
this  portion  of  the  solution.  The  systematic  study  of  this  phe- 
nomenon was  first  made  by  Faraday  in  1832.  He  called  the 
operation  of  separating  the  constituents  of  a  compound  in  solu- 
tion by  means  of  a  current  electrolysis.  Any  liquid  which  was 
capable  of  being  decomposed  in  this  way  he  called  an  electrolyte. 
The  constituents  of  the  dissolved  substance  which  appeared  at 
the  electrodes  he  called  ions.  The  results  of  his  investigation 
may  be  summarized  thus. 

(1)  So  long  as  the  deposit  of  but  one  particular  kind  of  ion, 
for  example   silver,   was    being    studied,   it   was   found   that   the 
amount  of  the  deposit  by  weight  was  proportional  solely  to   the 
quantity    of   electricity   which   had  passed   through    the   solution. 
It  was  independent  of  the  nature  of  the  solvent,  of  the  nature 
of  the  silver  compound  in  the  solution  (AgCl,  AgX03,  Agl,  etc.), 
of   the   concentration    of   the    solution,   and   of    the    strength   of 
the  current,  save  as  this  depended   upon  quantity  through  the 
relation   Q  =  It. 

The  obvious  interpretation  of  this  result  is  that  a  given  atom, 
or  radical,  in  whatever  compound  it  is  found,  is  always  associated 
with  a  certain  definite  quantity  of  electricity.  In  other  words, 
electricity,  like  matter,  consists  of  discrete  parts.  The  charge  asso- 
ciated with  an  atom  of  hydrogen  was  first  called  by  Helmholtz 
an  atom  of  electricity. 

(2)  When  the  investigation  was  extended  to  different  ions  of 
the  same  valency  (combining  power  with  respect  to  hydrogen), 
such,  for  example,  as  hydrogen,  silver,  iodine,  or  bromine,  it  was 


38  ELECTEICITY,  SOUKD,  AND  LIGHT 

found  that  the  amounts,  ~by  weight,  deposited  (or  liberated  in  the 
case  of  gases)  ly  the  passage  of  equal  quantities  of  electricity  were 
exactly  proportional  to  the  atomic  (or  ionic)  weights  of  the  ions. 
Thus  an  atom  of  silver  weighs  about  108  times  as  much  as  an 
atom  of  hydrogen,  and  the  passage  of  one  coulomb  of  electricity 
through  a  solution  containing  a  silver  salt  was  found  to  deposit 
108  times  as  many  grams  of  silver  as  there  were  grams  of  hydrogen 
liberated  by  the  passage  of  one  coulomb  of  electricity  through  a 
solution  of  hydrochloric  acid  (HC1)  in  water. 

This  evidently  means  that  an  atom  of  silver  carries  exactly  the 
same  charge  as  an  atom  of  hydrogen,  or,  in  general,  that  all  atoms 
having  the  same  valency  carry  the  same  charges  of  electricity. 

(3)  When  substances  of  different  valencies  were  compared,  as 
for  example  hydrogen  and  copper,  it  was  found  that  the  deposits 
ly  weight  produced  ly  the  passage  of  a  given  quantity  of  electricity 
were  proportional  to  the  atomic  weights  of  the  ions  divided  ly  their 
valencies.  Thus  the  copper  atom  has  a  valency  2  and  a  weight  of 
63.2  as  compared  with  the  atom  of  hydrogen.  The  deposit  of 
copper  due  to  the  passage  of  a  given  quantity  of  electricity  was 

/» q  o 

not  63.2  times  the  deposit  of  hydrogen,  but  instead  exactly  -  —  > 

2i 

or  36.6  times  this  deposit.  This  signifies  that  a  copper  atom  car- 
ries twice  as  large  a  charge  as  a  hydrogen  atom,  i.e.  that  it  carries 
two  of  Helmholtz's  atoms  of  electricity,  or,  in  general,  that  the 
chemical  valencies  1,  2,  3,  4>  5  correspond  to  electrical  charges 
upon  the  ions  in  the  ratios  1,  2,  3,  4->  &- 

These  discoveries  certainly  indicate  that  chemical  attractions 
are  of  electrical  origin.  They  constitute  the  chief  ground  for  the 
statement  made  in  section  24,  page  28,  that  the  passage  of  a  cur- 
rent through  a  liquid  consists  in  the  movement  through  the  liquid 
of  swarms  of  electrically  charged  particles  (the  ions). 

The  weight  of  a  substance  in  grams  deposited  by  the  passage  of 
one  absolute  electro-magnetic  unit  of  quantity  of  electricity  is 
called  the  electro-chemical  equivalent  of  that  substance.  Thus  the 
electro-chemical  equivalent  of  silver  (atomic  weight  107.7,  valency  1) 
is.01118,that  of  hydrogen  (atomic  weight  1,  valency  l)is. 0001038, 
that  of  copper  (atomic  weight  63.2,  valency  2)  is  .00328. 


MEASUREMENT  OF  ELECTRIC   CURRENTS          39 

32.  Voltameter.    This  .phenomenon  of  electrolysis  is  applied  to 
the    measurement    of   currents  in  an  instrument   known  as  the 
voltameter.    The  name  "  voltameter  "  is  applied  to  any  arrangement 
of  apparatus  whereby  the  weight  or  the  volume  of  the  ions  caused 
to  pass  out  of  an  electrolytic  solution  is  used  to  determine  the 
total  quantity  of  electricity  which  has  passed  through  the  circuit 
in  which  the  instrument  is  placed.    Because  of  the  accuracy  with 
which  weighings  can  be  made,  and  because  of  the  fact  that,  by 
taking  the  time  long  enough,  a  very  small  current  can  be  made 
to  deposit  a  considerable  weight  of  metal,  the  voltameter,  rather 
than  the  tangent  galvanometer,  is  the  instrument  which  is  almost 
universally  used  for  standardizing  current-measuring  instruments. 
However,  it  is  to  be  remembered  that  the  electro-chemical  equiva- 
lents had  first  to  be  determined  by  reference  to  the  tangent  gal- 
vanometer, or  some  similar  instrument  which  measures  directly 
the  strength  of  the  magnetic  field  at  a  known  distance  from  the 
current,  and  hence  that  the  fundamental  current-measuring  instru- 
ment is  not  the  voltameter,  but  the  galvanometer. 

33.  Ammeter.    Neither  the  tangent 
galvanometer   nor  the  voltameter   are 
convenient  for  ordinary  commercial  or 
laboratory  work.     Instruments   called 
ammeters  (see  Fig.  24)  are  universally 
used  for  measuring   current   strength. 
These   instruments   are  merely   galva- 
nometers, of  a  type  to  be  described  later, 
which  have  been  empirically  calibrated 

by  passing  a  given  current  simultaneously  through  the  ammeter 
and  through  an  electrolytic  solution  (voltameter). 

EXPERIMENT  3 

Object.  To  measure  a  current  by  means  of  (1)  two  copper  voltameters, 
(2)  a  silver  voltameter,  (3)  a  hydrogen  voltameter,  (4)  a  tangent  galvanom- 
eter, and  (5)  an  ammeter  ;  otherwise  stated,  to  test  Faraday's  laws  and 
to  calibrate  an  ammeter. 

Directions.  I.  The  solutions.  Make  up  two  solutions  of  pure  copper  sul- 
phate, one  containing,  say,  22  g.,  and  the  other  32  g.  of  crystals  to  100  g. 
of  water.  Add  to  each  solution  about  1  per  cent  of  strong  sulphuric  acid. 


40 


ELECTRICITY,  SOUND,  AND  LIGHT 


Make  up  for  the  silver  voltameter  a  neutral  silver  nitrate  (AgNO3)  solu- 
tion by  dissolving  about  18  g.  of  crystals  in  100  g.  of  water.  Also  make  up 
for  the  hydrogen  voltameter  a  solution  of  50  or  60  g.  of  sulphuric  acid  to  a 
liter  of  water. 

II.  Connections  and  adjustments.  Set  the  coil  of  the  tangent  galvanom- 
eter in  the  magnetic  meridian,  i.e.  parallel  to  the  needle  of  a  good 
compass.  Then  by  means  of  the  leveling  screws  adjust  the  instrument 
until  the  suspending  fiber  hangs  over  the  center  of  the  circular  scale. 
Finally  by  means  of  the  torsion  head  bring  the  suspended  needle  into  the 


magnetic  meridian.  This  should  be  accomplished  when  each  end  of  the 
aluminum  index,  which  is  attached  rigidly  to  the  needle  at  right  angles 
to  its  length,  is  exactly  above  one  of  the  two  zeros  of  the  scale. 

Connect  as  in  the  diagram  (Fig.  25),  taking  especial  care  to  avoid  loose 
contacts.  The  battery  B  may  consist  of  three  or  four  storage  cells  or  six 
or  eight  fresh  dry  cells.  The  order  in  which  various  instruments  are 
placed  in  the  circuit  is  wholly  unimportant,  but  the  tangent  galvanometer 
G  should  be  not  less  than  ten  feet  away  from  the  milliammeter  A  in  order 
that  the  magnet  in  the  latter  may  not  influence  the  needle  of  the  former. 
Furthermore,  since  the  deposit  of  a  metal  or  of  hydrogen  is  always  upon 


MEASUREMENT   OF  ELECTRIC   CURRENTS 


41 


FIG.  26 


the  plate  toward  which  the  current  flows  in  the  solution  (the  cathode), 

care  in  getting  the  direction  of  the  current  through  the  voltameters  as 

indicated  by  the  arrows  is  of  the  utmost  importance.    The  light  lines  in  B 

represent  the  +  terminals,  i.e.  the  carbons  of  dry  cells.    The  4-  terminals- 

of  storage  cells  are  usually  marked.     If  they 

are  not  marked  they  can  easily  be  determined 

by  connecting  the  two  terminals  with  a  few 

feet  of  about  No.  30  German  silver  wire 

and  noting  the  direction  in  which  a  compass 

needle  held  near  the  wire  is  deflected  (see 

right-handed-screw7  rule,    p.    32).     R    is    a 

cheap,    variable    German    silver    resistance 

provided  with  a  strong  spring  clamp  for  a 

sliding  contact  (Fig.  38,  p.  53).    J/  is  a  mercury  commutator  (Fig.  26),  so 

made  that  connecting  the  mercury  contacts  e/and  gh  causes  the  current  to 

traverse  the  tangent  galvanometer  G  in  one  direction,  while  connecting 

the  mercury  cups  eg  and  fh  causes  the 
current  to  pass  through  the  galvanometer 
in  the  opposite  direction.  The  negative 
electrodes  a  and  b  are  pure  copper  plates 
of  at  least  50  sq.  cm.  area  per  ampere  of 
current  and  well  rounded  at  the  corners  ; 
the  positive  electrodes  are  either  bent  cop- 
per plates  of  slightly  greater  width,  as  in 
Figure  25,  or  else  two  separate  plates  elec- 
trically connected  as  shown  in  Figure  27. 
The  silver  voltameter  has  similar  electrodes 
of  pure  silver,  the  gain  plate  having  an  area 
of  about  300  sq.  cm.  per  ampere.  In  the 
instrument  shown  in  Figure  27  the  plates 
slip  conveniently  into  spring  clips  attached 
to  the  lower  side  of  the  cover.  The  elec- 
trodes entering  the  graduated  tubes  0  and 
//consist  of  pieces  of  platinum  foil  attached 
to  platinum  wires.  Each  wire  may  be  sealed 
into  the  end  of  a  glass  tube  bent  into  a 
U -shape,  so  that  the  platinum  foil  may  be 
well  within,  the  graduated  tubes ;  or  the 

instrument  may  be  constructed  in  the  manner  shown  in  Figure  28. 

If  the  plates  have  been  in  recent  use  proceed  as   follows.*    Place  the 

plates  in  their  respective  clips  and  lower  them  into  the  solutions.    Fill  the 


FIG.  27 


*  If  the  gain  plates  are  not  already  clean  they  should  be  thoroughly  polished 
with  glass  or  sandpaper  (not  with  emery). 


42 


ELECTEICITY,  SOUND,  AND  LIGHT 


tubes  0  and  H  with  the  sulphuric  acid  solution.  Give  the  resistance  R 
its  maximum  value.  Close  the  circuit  and  then  adjust  R  until  the  deflec- 
tion of  the  tangent  galvanometer  needle  is  about  45°.  (The  time  occupied 
by  the  experiment  will  be  of  convenient  length  if  the  galvanometer  is  so 
wound  as  to  make  this  correspond  to  a  current  of  from  200  to  250  milli- 
amperes.)  Allow  the  current  to  flow  for  four  or  five  minutes  so  as  to  form 
a  fresh  deposit  upon  the  plates.  Then  remove  the  gain  plates,  rinse  first 
in  distilled  water,  then  in  alcohol,  and  finally  dry  over  a  radiator  or,  if 
time  is  limited,  by  igniting,  in  an  alcohol  or 
Bunsen  flame,  the  alcohol  which  cannot  be  shaken 
off.  Do  not  touch  the  deposit  at  any  time  with 
the  fingers.  ' 

III.  Weighings  and  readings.  AVeigh  the  plates 
as  follows.  By  means  of  a  wire  hook  hang  the 
plate  to  be  weighed  from  one  balance  arm  and 
add  weights  to  the  other  pan  until  the  resting 
point  of  the  pointer,  as  determined  by  the  method 
of  oscillations,*  is  within  one  or  two  divisions  of 
the  middle  of  the  scale  over  which  the  end  of  the 
pointer  moves.  Record  this  resting  point,  count- 
ing the  middle  division  on  the  scale  as  10,  and 
record  also  the  weight  which  you  have  placed  on 
the  other  pan.  Do  this  with  each  of  the  three 
cathodes,  then  replace  them  in  their  respective 
solutions,  f  Read  the  levels  of  the  liquid  in  the 
tubes  0  and  //.  Then,  at  some  accurately  ob- 
served time,  turn  on  the  current.  Allow  it  to 
run  until  40  or  50  cc.  of  hydrogen  have  been 
formed,  continually  adjusting  R,  if  necessary, 
so  that  the  ammeter  reading  is  kept  very  con- 
stant. While  the  current  is  flowing  read  both 
ends  of  the  tangent  galvanometer  index,  taking 
great  pains  in  so  doing  to  set  the  eye  directly 
over  the  center  of  the  scale.  At  any  convenient 
time  during  the  run  reverse  the  current  in  the 
FIG.  28  tangent  galvanometer  and  again  read  both  ends 

of   the    index.     In    reversing   leave    the    current 

open  long  enough  to  allow  the  needle  to  make  one  half  swing.  Other- 
wise the  needle  may  swing  completely  around.  This  interval  of  open 
circuit,  which  may  well  be  measured  with  a  stop  watch,  is  to  be  deducted 

*  See  "Mechanics,  Molecular  Physics,  and  Heat,"  Experiment  15. 

t  The  plates  should  not  be  placed  in  the  solutions  more  than  a  minute  or  two 
before  the  deposition  is  begun,  for  the  copper  plates  tend  to  dissolve  slowly 
in  the  solutions. 


MEASUKEMENT   OF  ELECTRIC   CURRENTS          43 

from  the  total  observed  time  in  order  to  obtain  the  actual  time  of  dura- 
tion of  the  experiment.  At  an  accurately  observed  time  break  the  circuit  ; 
remove  the  plates,  rinse  all  three  carefully  in  distilled  water,  then  in 
alcohol,  being  exceedingly  careful  not  to  shake  off  any  of  the  loosely 
adhering  silver  granules  ;  dry  over  a  radiator  or  by  igniting  the  film  of 
alcohol  which  clings  to  the  surface,  and  weigh  again  as  follows.  Placing 
each  plate  on  the  same  balance  pan  as  before,  record  the  weight  which 
will  bring  the  pointer  to  within  two  or  three  divisions  of  its  first  resting 
point.  Record  also  the  new  resting  point.  Add  2  mg.  and  record  the 
resting  point  for  this  added  weight.  From  these  last  two  resting  points 
determine  the  sensibility  of  the  balance,  i.e.  the  number  of  scale  divi- 
sions corresponding  to  an  addition  of  one  mg.  to  the  pan.  Using  this  sen- 
sibility determine  the  weight,  to  one  tenth  mg.,  which  wrould  bring  the 
resting  point  of  the  second  weighing  into  coincidence  with  that  found  in 
the  first  weighing.  Thus  suppose  that,  before  the  deposition,  with  a  weight 
on  the  right  pan  of  35.965  g.  the  resting  point  was  9.5,  and  that  after 
the  deposition  a  weight  of  36.195  g.  produced  a  resting  point  of  8.6, 

while  the  addition  of   2  mg.  to  the  left  side  changed  the  resting  point 

i  n  T      ft  f\ 
to  10.7.     The  sensibility  is   then  —  —=1.05,  and  the  number  of 

milligrams    corresponding    to  'the    difference    in    resting    points   between 

9.5  and  8.6  is  —  -  —  —  —  =  .9  mg.     Hence  the  wreight,  after  the  deposit, 
J.  .  (Jo 

which  would  have  been  necessary  to  produce  a  resting  point  of  9.5 
was  36.195  -  .0009  =  36.1941.  Hence  the  weight  of  the  deposit  was 
36.1941  -  35.965  =  .2291  g. 

IV.  Calculation  of  current.  The  calculation  of  the  quantity  of  electricity 
which  has  accompanied  the  deposits  of  silver  and  copper  is  easily  made 
from  these  deposits  and  the  electro-chemical  equivalents  given  on  page  38. 
The  current  is  then  obtained  from  equation  (1),  page  33.  The  calculation  of 
the  current  from  the  amount  of  hydrogen  collected  can  be  made  as  follows. 

Let  V  represent  the  observed  volume  of  hydrogen,  let  VQ  represent  this 
volume  reduced  to  0°C.  and  76cm.  pressure.  The  quantity  of  electricity 

which  has  passed  (measured  in  absolute  units)  is  equal  to  —  ^-,   since 

1.156  cc.  is  the  volume  at  0°C.  and  760  mm.  pressure  occupied  by 
.0001038  g.  (the  electro-chemical  equivalent)  of  hydrogen.  V0  is  obtained 
from  V  by  the  equation  representing  the  combination  of  the  laws  of  Boyle 
and  Charles,  namely 


V  P    T 

7 


<> 


-  ,  C6) 


V       760  T 

in  which  T  is  the  absolute  temperature  of  the  room  and  P  the  pressure 
which  the  hydrogen  in  the  tube  exerts.  If  B  represents  the  existing  barom- 
eter height  in  millimeters,  h  the  height,  also  in  millimeters,  of  the  final 


44  ELECTRICITY,  SOUND,  AND  LIGHT 

« 
level  of  the  water  in  the  tube  above  that  in  the  outer  vessel  (in  the  instru- 

ment of  Fig.  28  this  will  of  course  be  negative),  and  p  the  pressure  of 
saturated  water  vapor  (expressed  in  millimeters  of  mercury  at  the  tempera- 
ture of  the  room),*  then  evidently 


the  number  12  being  taken  as  the  approximate  ratio  between  the  density 
of  mercury  and  that  of  the  II2SO4  solution. 

In  the  calculation  of  the  current  from  the  readings  of  the  tangent  gal- 
vanometer a  sufficiently  accurate  correction  for  the  torsion  of  the  fiber  may 
be  made  without  determining  T0.  Thus,  while  no  current  is  flowing 
through  the  galvanometer,  twist  the  torsion  head  through  the  same  angle  6 
through  which  the  needle  was  deflected  when  the  current  was  flowing. 
Add  to  6  the  angle  through  which  the  needle  is  now  deflected  in  order  to 
obtain  a  corrected  value  of  0  for  substitution  in  equation  (5).  The  num- 
ber of  turns  of  the  galvanometer  will  be  given  by  the  instructor.  The 
value  of  H  has  .been  determined  in  Experiment  2,  (C).  Measure  the  mean 
diameter  of  the  coil.  Calculate  the  current  from  the  data  at  hand. 


EXAMPLE 

After  the  current  had  been  allowed  to  flow  for  several  minutes  the 
plates  were  rinsed,  dried,  and  weighed.  The  current  was  then  allowed  to 
flow  for  24  minutes  and  30  seconds,  less  7  seconds  required  for  the  reversal 
through  the  tangent  galvanometer,  i.e.  the  current  flowed  for  1463  seconds. 
The  record  of  the  weighing  was  as  follows.  With  the  first  copper  plate  the 
weight  used  was  20.190  g.  and  the  resting  point  11.75.  After  the  deposit 
20.290  g.  gave  a  resting  point  of  11.53.  Adding  2  mg.  gave  a  new  rest- 
ing point  of  9.00.  Hence  the  sensibility  was  1.26  and  the  correct  second 
weight  20.2898  g.  Hence  the  gain  was  .0998  g.  and  the  current  .2070 
ampere.  With  the  second  copper  plate  the  first  weight  was  19.695  g.  at 
12.75  and  the  second  19.795  at  11.75.  As  the  mass  was  practically  that 
of  the  first  plate  weighed,  it  was  unnecessary  to  redetermine  the  sensibil- 
ity; hence  the  second  weight  was  19. 7942  g.,  the  gain  .0992g.,  and  the 
current  .2058  ampere. 

Similarly  with  the  silver  plate  the  first  weight  was  20. 540  g.  at  11.56 
and  the  second  20.878  g.  at  11.30.  Hence  the  final  weight  was  20.8778  g., 
the  gain  .3378  g.,  and  the  current  .206"3  ampere. 

The  initial  volume  of  the  hydrogen  was  7.24  cc.  and  the  final  47.04cc. 
Hence  the  gain  was  39.8  cc.  The  barometer  reading  was  744  mm.  at 

*  See  table  No.  1  in  the  Appendix. 


MEASUREMENT  OF   ELECTRIC   CURRENTS          45 

22°  C.    This  reduced  to  0°C.  gave  B  —  741.4;   //  was  25  mm.  ;  p  from  the 
table  was  19.6  mm.    Hence  P  —  741.4  —  2.1  —  19.6  =  719.7.    Hence 

^  273       .39.8  =  34.8, 


700      273  +  22 

and  the  value  of  the  current  as  found  from  this  was  .2062  ampere. 

The  readings  on  the  tangent  galvanometer  were,  for  the  east  end,  47.  G, 
for  one  direction  of  the  current,  and  47.7  for  the  reversed  direction.  For 
the  west  end  these  readings  were  47.5  and  47.6,  respectively,  giving  a 
mean  of  47.6°.  Twisting  the  torsion  head  through  48°  caused  a  deflec- 
tion of  the  needle  of  .5°.  Hence  6  —  48.1°.  The  average  radius  of  the  coil 
was  15  cm.,  and  there  were  24  turns,  //"from  Experiment  2  was  .1856. 
Hence  the  current  was  .2053  ampere. 

The  ammeter  reading  was  kept  constant  throughout  the  experiment  at 
.207  ampere. 

The  average  of  all  the  determinations  of  current  was  .2059  ampere,  and 
the  greatest  deviation  of  any  single  determination  was  .3  per  cent.  The 
error  in  the  ammeter  at  this  point  was  approximately  1  milliampere. 


CHAPTEE  IV 
THE   MEASUREMENT   OF  POTENTIAL  DIFFERENCE 

34.  Potential  difference  between  points  in  the  neighborhood  of 
charged  bodies.  The  gravitational  P.D.  between  two  points  has 
been  denned  as  the  amount  of  work  required  to  carry  unit  mass 
between  the  two  points  against  the  force  of  the  existing  gravita- 
tional field.  An  exactly  analogous  definition  was  given  for  the 
electrical  P.D.  between  two  points,  viz.  the  number  of  ergs  of  work 

required    to    carry    unit    positive 
[     i     ]  A  B     charge    between    the    two    points 

\ /          FlG  29  against  the  force  of  the  existing 

electrical    field.     This    definition 

holds  under  all  circumstances,  whether  static  or  current  phe- 
nomena are  under  consideration.  Thus  the  P.D.  between  the 
points  A  and  B  (Fig.  29)  near  a  charge  P  is  simply  the  number 
of  ergs  of  work  required  to  carry 
unit  4-  charge  against  the  re-  f  ,  ^ 

pulsion  of  P  from  B  to  A ;  or,     \^^J  FJG  30  x^x 

inversely,  the*  work  which  the 

electric  field  does  in  carrying  unit  -f  charge  from  A  to  B.  Or 
again,  if  the  points  A  and  B  represent  the  localities  of  +  and  — 
charges  respectively  (see  Fig.  30),  the  P.D.  between  A  and  B  is  the 

number  of  ergs  of  work  required  to 
x  A  B       carry  unit  -f  charge  from  B  to  A. 

If  the  points  A  and  B  in  Fig- 
ure 29  are  connected  by  a  con- 
ductor of  any  sort,  as  in  Figure  31,  we  have  seen  (sect.  13)  that  a 
current  at  once  flows  through  the  conductor,  and  for  convenience 
we  have  taken  the  direction  of  this  current  the  same  as  the  direc- 
tion of  the  electric  force,  i.e.  from  A,  the  point  of  higher  potential, 
to  B,  the  point  of  lower  potential.  We  have  seen  further  that  in 
this  case  the  current  can  be  only  momentary,  for  since  AB  is  an 

46 


MEASUREMENT  OF  POTENTIAL  DIFFERENCE      47 

insulated  conductor,  the  passage  of  a  current  from  A  to  B  means 
the  appearance  of  a  +  charge  at  B,  and  of  a  —  charge  at  A,  since  + 
and  —  charges  always  appear  simultaneously  and  in  equal  amount. 
But  since  the  accumulation  of  a  +  charge  at  B  and  a  —  charge  at 
A  means  the  creation  of  a  field  of  force  between  A  and  B  whose 
direction  is  opposite  to  that  of  the  field  due  to  P,  we  saw  that  a 
current  can  flow  through  an  isolated  conductor  in  obedience  to 
the  force  of  an  outside  electric  field  only  until  the  new  field  be- 
tween A  and  B}  created  by  the  charges  accumulating  at  these 
points,  is  sufficiently  strong  to  neutralize  the  initial  field,  i.e.  the 
fieKTdue  to  P ;  that  is,  strong  enough  to  reduce  to  zero  the  force 
which  was  initially  causing  a  current  to  flow  from  A  to  B.  This 
force  gone,  there  can  of  course  be  no  longer  any  P.D.  between  A 
and  B ;  for  the  existence  of  a  P.D.  depends  upon  the  existence  of 
an  electric  force  (sects.  9  and  11).  It  was  thus  seen  that  as  soon 
as  a  current  ceases  to  flow  through  a  conductor  in  an  electric 
field,  all  of  the  points  of  the  conductor  must  have  the  same  poten- 
tial. Similarly,  in  Figure  30,  the  result  of  connecting  A  and  B  by 
a  conductor  is  to  cause  a  current  to  flow  between  these  points 
until  the  field  strength  existing  between  them  is  reduced  to  zero, 
i.e.  until  A  and  B  have  the  same  potential. 

35.  Potential  difference  between  points  on  the  terminals  of  an 
electric  generator.  But  if  A  and  B  are  the  terminals  of  a  static 
machine,  or  of  a  galvanic  cell,  or  of  a  dynamo,  or  of  any  generator 
E  (Fig.  32)  of  electricity  which  is 
capable  of  building  up  a  field  of 
given  strength  between  A  and  B, 
i.e.  of  maintaining,  on  open  circuit, 
a  given  P.D.  between  them,  then, 
when  these  points  are  connected 
by  a  conductor,  the  first  result 
must  be,  as  above,  that  a  current 

flows  from  A  to  B,  thus  tending  to  discharge  these  bodies,  and 
to  cause  the  field  between  them  to  collapse,  i.e.  to  cause  their 
P.D.  to  disappear.  But  the  generator  E  tends  instantly  to  re- 
charge A  and  B,  and  if  this  recharging  takes  place  at  a  rate 
which  is  very  rapid  in  comparison  with  the  rate  at  which  the 


48  ELECTBICITY,  SOUND,  AND  LIGHT 

connecting  conductor  discharges  them,  it  is  evident  that,  in 
spite  of  the  presence  of  the  conductor,  a  mean  strength  of  field 
may  be  kept  up  between  A  and  B  which  is  almost  as  great  as 
that  kept  up  when  the  connecting  conductor  was  absent.  On  the 
other  hand,  if  the  conductor  discharges  A  and  B  very  rapidly  in 
comparison  with  the  rate  at  which  the  generator  is  able  to  recharge 
them,  the  strength  of  field  maintained  between  them  must  be  small ; 
in  other  words,  A  and  B  will  have  nearly  the  same  potential.  Thus, 
connecting  the  points  A  and  B  by  a  very  good  conductor  may 
lower  their  P.D.  almost  to  zero,  —  never,  of  course,  quite  to  zero  so 
long  as  a  current  is  flowing  from  A  to  B,  —  while  connecting  them 
with  a  very  poor  conductor  may  lower  their  P.D.  an  inappreciable 
amount.  Connections  of  intermediate  conductivity  must  of  course 
produce  intermediate  lowerings  of  the  P.D.  between  A  and  B.  It  is 
evident,  then,  that  in  all  cases  a  P.D.  exists  between  two  points, 
ivhether  on  a  conductor  or  off  from  it,  only  by  virtue  of  the  existence  of 
a  static  field  which  makes  it  necessary  to  do  work  in  order  to  carry  a 
charge  from  one  point  to  the  other  against  the  existing  electrical  force. 
36.  Work  done  independent  of  path.  If  A  and  B  are  the  ter- 
minals of  a  cell  or  a  dynamo,  and  if  these  terminals  are  connected 
by  a  long  wire  ACDB  (see  Fig.  33),  the  assertion  that  a  certain 
P.D.,  say  10  absolute  units,  exists  between  A  and  B  means  simply 
that  it  would  require  10  ergs  of  work  to  carry  a  unit  +  charge 

straight  across  through  the 
air  against  the  force  of  the 
electric  field  which  exists 
between  A  and  B.  It  also 
means  that  it  would  re- 
quire just  the  same  work 
D  to  carry  a  unit  charge  from 

33  B  to  A  by  any  other  path 

whatever;    for   the    work 

done  is  wholly  independent  of  the  path  (see  sect.  12,  p.  11).  Or, 
conversely,  it  means  that  the  field  would  do  10  ergs  of  work  in  driv- 
ing a  pith  ball  charged. with  1  -+-  unit  from  A  over  to  B;  or,  since 
work  is  independent  of  path,  that  it  would  do  10  ergs  of  work  in 
carrying  one  unit  of  charge  from  A  to  B  through  the  wire  A  CDB. 


MEASUREMENT  OF  POTENTIAL  DIFFERENCE      49 


37.  How  the  work    appears.    If   the   pith  ball  were   driven 
through  the  air  from  A  to  B,  the  work  done  by  the  field  would 
be  expended  in  overcoming  the  friction  of  the  air  and  in  impart- 
ing kinetic  energy  to  the  ball.    This  kinetic  energy  would  all  be 
given  up,  and  a  corresponding  heat  energy  would  appear,  when 
the  pith  ball  struck  B.    Hence  all  of  the  work  done  in  moving 
the  +  unit  charge  from  A  to  B  would  appear  as  heat.    Similarly 
if  the  unit  charge  goes  from  A  to  B  through  the  wire  ACDB,  the 
10  ergs  of  work  expended  by  the  field  in  carrying  it  in  this  way 
from  A  to  B  must  in  general  appear  as  heat.    If  any  chemical 
changes  took  place  in  the  wire  because  of  this  passage  of  current, 
this  conclusion  would  not  hold,  nor  would  it  hold  if  an  electric 
motor  were  interposed  anywhere  in  the  wire  ACDB.    In  these 
cases  the  work  done  by  •  the  field  would  be  equal  to  the  heat 
developed  plus  the  chemical  or  mechanical  work  done.     But  if 
the  heat  is  the  only  observable  effect  produced,  then  it  follows 
from  the  principle  of  the  conservation  of  energy  that  the  work 
done  by  the  electric  field  in  moving  a  unit  charge  from  A  to  B 
must  be  equal  to  the  heat  developed  in  the  wire  ACDB  when 
this  heat  has  been  reduced  to  mechanical  units.    According  to  the 
electron  theory  the  heating  is  in  this  case  due  to  the  frictional 
resistance   which    the 

wire  offers  to  the  pas- 
sage of  the  electrons 
through  it. 

38.  Measurement  of 
P.D.  It  is  evident  then 
from  the  definition  of 
P.D.  that  any  measure- 
ment of  P.D.  must  con- 
sist in  measuring  the 
work  done  in  carrying 
unit    charge    between 

the  two  points  whose  FlG  34 

P.D.  is  sought.     This 

work , may  be  obtained  by  measuring  the  strength  of  the  field  in 

dynes,  and  the  distance  between  the  points  whose  P.D.  is  sought 


50 


ELECTRICITY,  SOUND,  AND  LIGHT 


in  centimeters.  These  are  the  actual  measurements  which  are 
made  in  the  use  of  the  so-called  absolute  electrometer.  This  is  an 
instrument  which  consists  essentially  of  two  plates,  A  and  B  (Fig. 
34),  which  are  connected  by  wires  to  the  points  a  and  b  between 
which  the  P.D.  is  sought.  These  plates  are  so  .arranged  that  the 
force  of  attraction  between  them  may  be  measured.  From  the  force 
of  attraction  of  the  plates  the  strength  of  the  field  between  A  and 
B  is  easily  obtained.  The  product  of  this  field  strength  by  the 
distance  between  the  plates  gives  the  work  necessary  to  carry 
unit  charge  from  B  across  to  A.  Since  all  points  on  a  conductor 
in  which  no  current  flows  have  the  same  potential  (sect.  13,  p.  12), 
it  follows  that  the  potential  at  A  is  the  same  as  that  at  a,  and 
that  at  B  the  same  as  that  at  &,  i.e.  the  P.D.  between  A  and  B 
must  also  be  the  P.D.  between  a  and  b.  Commercial 
electrostatic  voltmeters,  which  are  coming  more  and 
more  into  use,  and  which  are  in  many  respects  the 
most  satisfactory  of  all  instruments  for  measuring 
P.D.,  differ  but  little  in  principle  from  the  absolute 
electrometer.  They  consist  of  two  fixed  plates  A 
and  B  (Fig.  35),  which  may  be  connected  to  the 
two  points  whose  P.D.  is  sought,  and  which  carry 
between  them  a  gold  leaf  g,  or  some  other  movable  system,  the 
deflections  of  which  may  be  taken  as  a  measure  of  the  field  strength 
between  the  plates.  The  instrument  is  empirically  calibrated  so 
that  given  deflections  of  the  movable  system  correspond  to  given 
differences  in  potential  between  A 
and  B,  and  therefore  between  the 
points  to  which  A  and  B  are  attached. 
Another  method  of  making  an 
absolute  measurement  of  P.D.  con- 
sists in  measuring  the  number  of 
calories  of  heat  developed  by  the 
passage  of  a  known  quantity  of  elec- 
tricity between  the  points  whose 
P.D.  is  sought.  Thus  suppose  that 

a  constant  current  of  strength  /  is  made  to  flow  through  a  platinum 
wire  AB  (Fig.  36)  immersed  in  a  calorimeter,  and  that  a  number  of 


FIG.  35 


FIG.  36 


MEASUREMENT   OF  POTENTIAL  DIFFERENCE      51 


calories  of  heat  M  are  thus  communicated  to  the  water  in  t  seconds. 
Then  the  amount  of  work  done  by  the  current  is  MJ  ergs,  where 
J  is  the  mechanical  equivalent  of  heat,  i.e.  the  number  of  ergs 
equivalent  to  one  calorie,  namely  4.19  x  107.  But  since  the  P.D. 
between  A  and  B  is  the  number  of  ergs  of  work  done  in  carrying 
unit  quantity  between  these  points,  the  work  done  by  the  passage 
of  Q(=  It)  units  of  quantity  may  be  written  PD  x  Q  =  PD  x  It. 
Equating  these  two  expressions,  we  obtain 

PD  It  =  MJ,       or       PD  =  —  '  (I) 

It 

Now  the  absolute  electro-magnetic  unit  of  P.D.  is  the  P.D. 
which  exists  between  two  points  when  it  requires  1  erg  of  work 
to  carry  1  unit  of  quantity  (measured  in  the  electro-magnetic 
system)  between  the  two  points.  The  above  equation  evidently 
gives  the  P.D.  in  such  units,  provided  M  is  measured  in  calories, 
t  in  seconds,  and  /  in  absolute  electro-magnetic  units  of  current. 
This  absolute  unit  of  P.D.  is  so  extremely  small,  however,  that  it 
has  been  decided  to  use  as  the  unit  in  practical  work  a  P.D. 
which  is  108  absolute  units.  This  unit  is  called  the  volt.  Hence 
the  P.D.  between  two  points  is  1  volt  when  it  requires  10s  ergs  of 
work  to  carry  one  electro-magnetic  unit  of  charge  between  these  points. 

39.  Voltmeters.  Neither  the  calorimetric  method  nor  that 
which  makes  use  of  the 
absolute  electrometer  is 
convenient  for  the  rapid 
measurement  of  P.D. ;  but 
either  one  of  them  may 
be  used  for  calibrating  a 
so-called  high-resistance 
galvanometer  in  such  a 
way  that  P.D.  may  be 
read  off  upon  it  directly. 
The  operation'  of  finding 
the  P.D.  between  any 

two  points  between  which  a  current  is  flowing  will  then  consist 
simply  in  touching  the  galvanometer  terminals  to  the  two  points 
whose  P.D.  is  sought,  and  observing  the  deflection  produced. 


FIG.  37 


52  ELECTRICITY,  SOUND,  AND  LIGHT 

Thus  suppose  a  given  P.D.  is  maintained  between  the  terminals 
C  and  D  of  a  platinum  wire  immersed  in  the  calorimeter  (Fig.  37) 
and  suppose  this  P.D.  has  been  found  by  measuring  the  rate  at 
which  heat  is  developed  in  the  calorimeter.  If  now,  while  the 
current  is  still  flowing,  a  galvanometer  be  connected  across  C 
and  D,  the  conductivity  of  the  galvanometer,  added  to  that  of  the 
wire  CD,  will  in  general  discharge  the  terminals  C  and  D  faster 
than  they  were  being  discharged  when  they  were  connected  by 
the  platinum  wire  alone.  Hence  the  introduction  of  the  galva- 
nometer will  in  general  cause  a  fall  in  the  P.D.  between  C  and  D. 
But  if  this  galvanometer  carries  a  quantity  ef  electricity  per  sec- 
ond which  is  wholly  negligible  in  comparison  with  that  carried 
by  CD,  then  the  P.D.  will  remain  essentially  unchanged  by  the 
introduction  of  the  galvanometer.  The  deflection  of  the  galvanom- 
eter may  then  be  marked  and  labeled  with  the  number  of  units 
of  P.D.  which  exist  between  C  and  D,  as  measured  by  the  heat 
developed  in  the  calorimeter.  The  P.D.  between  C  and  D  may 
then  be  increased  by'  the  use,  say,  of  a  different  generator,  and 
another  deflection  labeled  with  a  new  P.D.,  as  measured  by  the 
calorimeter.  In  this  way  the  galvanometer  may  be  empirically 
calibrated  throughout  its  whole  range  of  deflection,  so  that  its 
readings  always  indicate  the  P.D.  existing  between  the  points  to 
which  its  terminals  are  touched. 

It  is  evident  that  such  a  voltmeter  could  not  be  used  for 
determining  the  P.D.  between  two  isolated  static  charges,  for  the 
P.D.  between  these  charges  would  instantly  become  zero  as  soon 
as  the  galvanometer  terminals  were  touched  to  them.  It  could 
be  used  for  determining  the  P.D.  between  the  terminals  of  a 
dynamo,  or  of  a  galvanic  cell  on  open  circuit,  provided  it  carried 
so  little  current  as  not  to  lower  appreciably  this  P.D.  by  its  intro- 
duction between  them.  It  could  be  used  for  determining  the  P.D. 
between  two  points  on  a  conductor  which  is  already  carrying  a 
current,  provided  it  did  not  alter  this  P.D.  by  its  introduction  in 
shunt  with  the  conductor,  i.e.  provided  it  carried  a  current  which 
is  negligible  in  comparison  with  that  carried  by  the  conductor. 
Of  course  after  the  galvanometer  has  once  been  calibrated  for  P.D. 
its  readings  represent  under  all  circumstances  the  P.D.  existing 


MEASUREMENT   OF  POTENTIAL   DIFFERENCE      53 

between  its  own  terminals,  whether  the  current  which  it  carries 
is  negligible  or  not;  but  if  its  current  is  not  negligible,  the  P.D. 
between  the  points  to  which  it  is  touched  is  not  the  same  when  it 
is  in  contact  with  these  points  as  when  it  is  not  in  contact  with 
them.  Thus  a  voltmeter  is  merely  a  galvanometer  which  has  been 
calibrated  empirically  by  reference  to  some  absolute  measurement 
of  P.D. 

(Because  of  the  uncertainties  of  calorimetric  measurement,  the 
method  given  above  for  calibrating  a  voltmeter  is  not  the  one  which 
is  most  commonly  used  (see  Chap.  V)). 


EXPERIMENT  4 

Object.  To  test  the  calibration  of  a  voltmeter  by  an  absolute  determina- 
tion of  P.D. ;  otherwise  stated,  to  verify  the  relation  PD  x  Q  =  MJ. 

Directions.  Connect  as  in  the  diagram  of  Figure  37.  B  is  a  generator 
capable  of  producing  a  constant  P.D.  of  at  least  20  volts.  (Either  55  or 
110  volts  will  serve  the  purpose  admira- 
bly.)  A  is  an  ammeter  with  a  range  of, 
say,  0  to  10  amperes.  al>  is  a  platinum 
wire  (Xo.  30  to  No.  34)  which  is  capable 
of  being  raised  to  a  red  heat  by  a  cur- 
rent of  from  3  to  6  amperes.  S  is  a 
switch  which  is  to  be  left  open  until  the 
instructor  has  seen  that  all  connections 
have  been  properly  made,  and  which  is 
in  no  case  to  be  closed  unless  the  plati- 
num wire  is  immersed  in  water.  It  is  a 
variable  German  silver  resistance  (Fig. 

38)  capable  of  reducing  the  current  from  110  volt  mains  to  from  3  to  6 
amperes.  N  is  a  calorimeter  the  inner  vessel  of  which  has  a  capacity  of 
about  300  cc.  (Fig.  39  shows  cut  of  the  complete  calorimetric  outfit.) 
Figure  40  shows  the  construction  of  the  voltmeter  V  which  is  to  be 
calibrated. 

Fill  the  calorimeter  to  within  about  1  cm.  of  the  top  with  water  the 
temperature  of  which  is  from  10° C.  to  15°C.  below  the  temperature  of 
the  room,  provided  this  temperature  is  not  below  the  dew-point.  If  dew 
collects  on  the  vessel  the  initial  temperature  should  be  taken  a  degree  or 
two  above  that  at  which  this  dew  is  seen  to  appear.  Take  a  preliminary 
observation  as  follows.  Stir  thoroughly,  read  the  temperature,  then  throw 
on  the  current,  and  keep  it  on  for  just  one  minute,  the  ammeter  reading 


ELECTRICITY,  SOUND,  AND  LIGHT 


being  adjusted  to  3}  or  4  amperes.  If  the  final  temperature  is  not  more 
than  two  or  three  degrees  above  room  temperature,  the  conditions  need  not 
be  changed.  Otherwise  alter  the  current  until  a  run  of  a  minute  produces 
about  the  final  temperature  indicated  above.  Then  fill  the  calorimeter 
again  with  cold  water,  stir  it  thoroughly,  insert  the  platinum  coil,  and, 
^  while  stirring  continuously,  take  four  or  five 

successive  readings  of  the  temperature  of  the 
water,  estimating  the  thermometer  to  tenths 
of  the  smallest  division.  As  soon  as  possible 
after  the  last  reading  throw  on  the  current, 
and  keep  it  on  for  just  a  minute,  stirring 
vigorously  all  the  time  and  adjusting  R,  if 
necessary,  so  as  to  hold  the  current  constant. 
Stir  for  at  least  a  minute  after  the  current 
has  been  turned  off,  then  take  the  final  tem- 
perature very  carefully.  If  the  experiment  is 
performed  in  this  way,  no  correction  for 
radiation  will  in  general  be  necessary. 

But  if  the  duration  of  the  heating  is  several 
minutes,  or  if  the  final  temperature  is  8°C.  or 
more  above  room  temperature,  the  final  tem- 
perature corrected  for  radiation  may  be  found 
as  follows.  Observe  the  initial  temperature 
precisely  as  indicated  above ;  take  the  time  at 
which  the  heating  begins,  the  time  at  which 
it  ends ;  then,  beginning  1  minute  after  this 

^^^^=^r=^^^^^^      latter  time,  record  the  temperature  at  minute 
~~~~  Tfl  intervals  for   at  least  5  minutes,  the  stirring 

(I  being  continuous  from  the  instant  of  throwing 

on  the  current.  Next  plot  times,  measured 
from  this  instant,  as  abscissas,  and  tempera- 
tures as  ordinates,  in  the  manner  indicated  in 
Figure  41.  The  full  line  drawn  through  these 
points  shows  the  rate  of  cooling  by  radiation, 
and  this  line,  extended  back  so  as  to  cut  the 
vertical  line  drawn  through  the  time  at  which 
the  current  was  discontinued,  gives  the  tem- 
perature of  the  water  at  the  instant  of  cessation 
of  the  current.  This  quantity  could  not  have 
been  observed  directly  because  it  is  impossible,  however  vigorous  the  stir- 
ring, to  keep  all  parts  of  the  water  at  the  same  temperature  during  the 
heating.  Next  find  from  the  curve  the  loss  in  temperature  per  minute  per 
degree  above  room  temperature.  This  quantity,  multiplied  by  the  time  in 
minutes  during  which  the  current  was  flowing,  and  by  the  number  of 


FIG.  39 


MEASUREMENT   OF  POTENTIAL   DIFFERENCE      55 

degrees  that  the  average  temperature  during  this  time  (as  found  by  taking 
the  mean  of  initial  and  final  temperatures)  was  above  the  room  tempera- 
ture, gives  the  number  of  degrees  to  be  added  to  the  final  temperature  in 
order  to  obtain  the  final  temperature  corrected  for  radiation.  If  the  aver- 
age temperature  during  the  heating  is  below  room  temperature,  obviously 
heat  is  gained,  arid  the  correction,  found  exactly  as  indicated  above,  is  to 
be  subtracted. 

As  soon  as  possible  after  the  observations  on  temperature,  weigh  on  a 
rough  balance  the  inner  vessel  of  the  calorimeter  and  the  contained  water, 


FIG.  40 

then  the  empty  calorimeter,  then  the  stirrer.  Estimate  the  volume  of 
the  electrodes  in  cubic  centimeters  from  rough  measurement  of  their 
dimensions. 

The  water  equivalent  of  the  inner  vessel  of  the  calorimeter,  that  is, 
the  number  of  grams  of  water  which  would  be  raised  1°C.  by  the  num- 
ber of  calories  which  is  required  to  raise  the  calorimeter  1°C.  is  evi- 
dently the  weight  of  the  calorimeter  x  the  specific  heat  of  brass  (.094). 
The  water  equivalent  of  the  electrodes  is  their  volume  x  their  density 


56  ELECTRICITY,  SOUND,  AND  LIGHT 

(=  8.4)  x  .094.  Similarly  the  water  equivalent  of  the  stirrer  is  obtained 
from  its  weight  and  specific  heat.  You  can  probably  neglect  the  water 
equivalent  of  the  thermometer.*  The  total  weight  of  water  plus  these 
water  equivalents,  multiplied  by  the  observed  rise  in  temperature,  evidently 
gives  the  number  of  calories  of  heat  developed  during  the  experiment. 
One  calorie  is  equivalent  to  427  gram  meters  of  work  or  to  4.19  x  107  ergs. 
If  I  is  the  mean  current  in  amperes,  the  heat  developed,  wThen  reduced  to 
e"rgs,  represents  the  work  in  absolute  units  which  has  been  done  in  trans- 
ferring Q(=  7^/10)  absolute  units  of  electricity  from  C  to  D.  Hence  the 
P.D.,  or  the  work  expended  in  transferring  one  unit,  is  at  once  obtained. 
This  is  reduced  to  volts  by  dividing  by  108.  t 


EXAMPLE 

The  room  temperature  was  found  to  be  17.2°C.    Initial  temperature  of 
water  was  15.4°C.     Final  temperature  (see  Fig.  41)  was  29.0°C.     Mean 

29  +  15  4 
temperature  of  water  during  heating  was —    '- =  22.2°C.    Mean  rate 

of  cooling  at  end  of  experiment  (eleven  degrees  above  room  temperature) 
was  .15°  per  minute.  .'.  Mean  rate  of  cooling  at  one  degree  above  room 
temperature  would  be  .014°  per  minute.  .'.  Rate  of  cooling  at  22.2°  was 
.014°  x  (22.2  —  17. 2)  =  . 07°  per  minute.  .'.Final  temperature  corrected 
for  radiation  was  29.07°C.  .'.  Rise  in  temperature  was  13.67°C.  Weight 
of  water  and  calorimeter  was  282.9  g.  Weight  of  calorimeter  was  54.9  g. 
Weight  of  stirrer,  7.87  g.  .'.  Weight  of  water  was  228.0  g.  Water  equiv- 
alent of  calorimeter  was  5.16  g.  Water  equivalent  of  stirrer,  .74  g.  Vol- 
ume of  electrodes,  .64  cc.  Water  equivalent  of  electrodes,  .51  g.  Water 


*  To  calculate  the  water  equivalent  of  the  thermometer  note  that  the  heat 
capacity  of  mercury  per  unit  of  volume,  viz.  13.6  (  =  sp.  gr.)  x  .033(=sp.ht.)  =  .47, 
is  nearly  the  same  as  that  of  glass,  viz.  2.5(=sp.  gr.)  x.!9(=sp.  lit.)  =.45. 
Hence  to  calculate  the  heat  capacity  of  a  thermometer,  estimate  the  volume 
immersed,  and  multiply  by  .46. 

t  The  calculation  may  be  freed  from  the  use  of  powers  of  10  as  follows. 
The  practical  unit  of  work,  the  joule,  is  107  ergs.  One  calorie  is  therefore  4.19 
joules.  If,  then,  M  represents  the  number  of  calories  developed,  4. 19  M  is  the 
heat  developed  expressed  in  joules.  If  PD  and  I  are  expressed  in  volts  and 
amperes  respectively,  then  the  work  done  is  (PD)  108  I  lQ~lt  ergs  =  (PD)  It 
joules.  This  gives  (PD)  It  =  4.19  3f,  where  PD  and  I  are  expressed  in 
practical  units.  A  coulomb  is  defined  as  the  quantity  carried  by  a  current 
of  1  ampere  in  1  second.  In  general,  then,  the  product  of  volts  and 
coulombs  gives  joules.  Also,  since  the  practical  unit  of  power,  the  watt, 
is  defined  as  1  joule  per  second,  the  product  of  volts  and  amperes  gives  the 
rate  of  work  or  power  in  watts. 


MEASUREMENT  OF  POTENTIAL   DIFFERENCE      57 

equivalent  of  thermometer  was  .32  g.  . ' .  Total  water  equivalent  was  234.7  g. 
Duration  of  heating  was  60  seconds.  Mean  current  /,  4.08  amperes.  There- 
fore P.D.  was  54.90  volts.  The  reading  of  the  voltmeter  was  54.8,  which 


FIG.  41 


differs  from  the  calculated  P.D.  by  but  .2%.     The  voltmeter  was  therefore 
correct  within  the  limits  of  observational  error. 


CHAPTEE  V 
THE  MEASUREMENT  OF  RESISTANCE 

40.  Ohm's    law.    In    1829    the    Berlin   physicist    G.  S.  Ohm 
announced  the  famous  law  now  called  after  its  discoverer  Ohm's 
law.    This  law  asserts  that  when  a  continuous  current  is  flowing 
through  a  given  conductor,  the  temperature  of  which  is  kept  con- 
stant, the  ratio  of  the  P.D.  existing  between  the  terminals  of  the 
conductor  and  the  current  carried  by  the  conductor  is  a  constant, 
no  matter  what  the  value  of  the  current  may  be.    Symbolically, 

PD  PD 

=  constant,        or        —  R.  (1) 

It  is  customary,  as  above,  to  denote  this  constant,  the  value  of 
which  depends  only  upon  the  nature  of  the  conductor,  by  the  let- 
ter R,  and  to  name  it  the  electrical  resistance  of  the  conductor.  The 
name  is  natural,  because  when  the  value  of  R  is  large,  the  amount 
of  electricity  which  flows  per  second  through  the  conductor  as 
the  result  of  a  given  P.D.  is  small,  and  vice  versa.  Ohm's  law  is 
to  be  regarded  as  an  empirical  law,  which  may  be  verified  directly 
by  varying  the  value  of  the  P.D.  between  the  terminals  of  a  given 
conductor  and  measuring  the  corresponding  currents.  The  most 
convincing  evidence  for  the  rigorous  correctness  of  the  law  is 
found,  however,  in  the  agreement  which  exists  between  precise 
observations,  like  those  made  with  a  Wheatstone's  bridge,  and  cal- 
culations which  depend  upon  the  assumption  of  this  law.  So  far 
as  the  most  delicate  measurements  have  been  able  to  show,  Ohm's 
law  is  not,  like  Boyle's  law,  an  approximation,  but  is,  rather,  a  law 
of  rigorous  exactness. 

41.  Units    of  resistance.    The    absolute   unit   of  resistance  is 
defined  as  the  resistance  of  a  conductor  which  conveys  one  absolute 
electro-magnetic  unit  of  current  when  its  terminals  are  maintained 
at  a  P.D.  of  one  absolute  electro-magnetic  unit.    The  practical  unit 

58 


MEASUREMENT  OF  RESISTANCE  59 

of  resistance  is  the  resistance  of  a  conductor  which  carries  one 
practical  unit  of  current  (1  ampere)  when  its  terminals  are  main- 
tained at  a  P.D.  of  one  practical  unit  (1  volt).  This  practical  unit 
of  resistance  is  called  the  ohm.  Thus 

volts 

=  ohms. 


amperes 

Since  1  volt  =  108  absolute  units  of  P.D.,  and  1  ampere  =  10"1 
absolute  units  of  current,  it  is  evident  that  1  ohm  =  109  absolute 
units  of  resistance. 

It  appears  at  once  from  the  definition  of  R,  namely  PZ>//,  that 
the  resistance  of  a  conductor  is  determined  once  for  all  when  an 
absolute  measurement  has  been  made  both  of  the  P.D.  existing  at 
any  time  between  its  terminals  and  of  the  corresponding  current. 
Thus  the  resistance  of  the  conductor  ab  in  the  preceding  experi- 
ment is  known,  at  the  mean  temperature  of  the  calorimeter,  from 
the  measurements  there  made  upon  PD  and  /.  When  such  an 
absolute  measurement  of  the  resistance  of  a  wire  has  been  once 
made,  it  is  very  easy  to  preserve  the  wire  so  that  no  change  takes 
place  in  its  character,  and  it  is  also  easy  to  reproduce  the  temper- 
ature at  which  its  resistance  was  determined.  Hence  with  the  aid 
of  this  wire  and  an  ammeter  standardized  by  comparison  with  a 
silver  voltameter,  it  is  easy  to  standardize  any  voltmeter;  for 
according  to  Ohm's  law  the  P.D.  between  the  terminals  of  the  wire 
will  always  be  the  product  of  the  resistance  of  the  wire  times  the 
current  flowing  through  it,  so  that,  R  being  known,  we  have  only 
to  vary  the  current  /  in  a  known  way  and  take  in  every  case  the 
product  RI  in  order  to  obtain  PD.  This  is  a  method  actually 
used  in  standardizing  laboratories  for  calibrating  voltmeters. 

When  one  standard  resistance  has  been  found  by  any  absolute 
method,  other  standards  can  easily  be  obtained  by  the  method  of 
comparison  to  be  described  in  section  43.  Many  elaborate  investi- 
gations have  been  undertaken  for  determining  accurately  the  length 
of  a  mercury  column  1  sq.  mm.  in  cross  section  which  will  carry 
exactly  1  ampere  of  current  when  its  terminals  are  maintained  at 
a'  P.D.  of  1  volt,  that  is,  the  length  of  a  mercury  column  1  sq.  mm. 
in  area  which  has  1  ohm  of  resistance  according  to  the  above 


60  ELECTRICITY,  SOUND,  AND  LIGHT 

definition.  The  methods  employed  for  measuring  the  P.D.  have 
not  generally  been  calorimetric  methods  like  the  above,  but  have 
involved  electro-magnetic  principles  to  be  considered  later.  The 
mean  of  the  results  of  the  most  eminent  observers  gives  1  ohm 
as  the  resistance  at  0°C.  of  106.3  cm.  of  mercury  1  sq.  mm.  in 
cross  section.  Hence,  at  the  Electrical  Congress  in  Chicago  in 
1893,  the  resistance  of  a  mercury  column  1  sq.  mm.  in  area  and 
106.3cm.  long  was  adopted  as  the  "international  ohm."  Stand- 
ards of  resistance  are  made  by  comparison  with  the  resistance  of 
such  a  mercury  column. 

42.  Laws  of  resistance.  Experiment  shows  that  R  varies  with 
the  length,  the  area,  the  temperature,  and  the  material  of  the  con- 
ductor. It  is  found  to  be  directly  proportional  to  the  length  and 
inversely  proportional  to  the  sectional  area.  For  metallic  con- 

d       ductors  it  is  found  to  in- 
^\   crease  with  the  tempera- 
ture,   the    coefficient    of 
1  increase  per  degree  being 

nearly  the  same  for  all 
FIG.  42  J 

pure  metals. 

If  the  separate  resistances  of  conductors  connected  in  series 
are  Rv  R2,  and  R3  (see  Fig.  42),  the  joint  resistance  of  the 
three,  that  is,  the  total  resistance  R  between  the  points  a  and 
d  is  easily  proved  to  be 

Zr  (2) 


For,  if  PD:  represents  the  potential  difference  between  a  and  b, 
PDZ  that  between  b  and  c,  PDS  that  between  c  and  d,  and  PD 
that  between  a  and  d,  then,  since  the  current  which  is  flowing  in 
all  parts  of  the  circuit  must  be  the  same,  we  have 

^.  (3) 


But  PD  =  PDl  +  PD2  +  PDS.  (4) 

(5) 

(6) 


PD      PD       PD0  , 

Hence  —     _i  +  _*  +  _».  (5) 


MEASUREMENT   OF  RESISTANCE 


61 


If  the   conductors   are  connected  in  parallel  (see  Fig.  43),  it 
can  easily  be  shown  .that  the  joint 
resistance  R,  i.e.  the  total  resist- 
ance between  the  points  a  and  I, 
is  given  by  the  equation 


RI      R2      R3 


For,  if  PD  represents  the  potential 
difference  between  the  points  a 
and  b}  then,  by  Ohm's  law, 

PD  PD  PD 


"3*3 


FIG.  43 


R  Rl  R2 


But  the  total  current  /  between  a  and  b  must  be  the  sum  of 
Iv  J2,  and  Iy 

PD      PD      PD      PD 

Hence  -_  =  _.  +  _.  +  _..  (8) 


PD      PD      PD 
R*        R~       Rn 


R 


(9) 


43.  Theory  of  the  Wheatstone  bridge.     Since  at  the  present 
time  standards  of  resistance  are  obtained  not  from  the  absolute 

measurement  of  P.D. 
and  current,  but  from 
comparison  with 
standards  already 
existing,  a  very  accu- 
rate method  of  com- 
paring resistances  is 
of  extreme  impor- 
tance. Wheatstone '&. 
bridge  is  an  instru- 
ment  By  means  of 
which  such  accurate 
comparisons  can  be 
made.  It  consists  of  a  divided  circuit  abc  and  adc  (Fig.  44)  com- 
posed of  three  known  resistances  P,  Q,  R,  and  of  the  unknown 


FIG.  44 


62  ELECTRICITY,  SOUND,  AND  LIGHT 

resistance  X,  the  value  of  which  is  to  be  found.  The  resistances 
P  and  Q  are  in  general  the  parts  ab  and  be  respectively  of  a  long 
German  silver  wire  connecting  a  and  c. 

As  soon  as  the  current  has  become  constant  in  the  circuit,  a 

definite  P.D.  must  exist  between  the  points  a  and  c.    Hence  the 

\total  fall  of   potential  along  the  branch  abc  must  be  the  same 

as  the  fall  of  potential  along  the  branch  adc.    It  is  evident,  there- 

fore, that  there  must  exist  somewhere  on  abc  a  point  which  has 

exactly  the  same  potential  as  the  point  d  on  adc.    This  point  can 

/easily  be  found  by  connecting  one  terminal  of  a  galvanometer  at  d, 

/and  sliding  the  other  terminal  along  abc  until  the  point  b  of  no 

J  deflection  is  reached.    This  must  be  the  point  which  has  the  same 

!  potential  as  the  point  d.    When  the  bridge  has  been  put  into  this 


condition  of  balance,  let  PD^  be  the  difference  of  potential  between 
a  and  d,  PD^  that  between  d  and  c,  PDS  that  between  a  and  b, 
and  PD4  that  between  b  and  c.  Also  let  /x  and  /2  be  the  currents 
flowing  in  the  upper  and  lower  branches  respectively.  Then,  by 
Ohm's  law, 

PDl  PD2 

~     =         and       -    =    ' 


„  PI).       PD,  R       PD. 

Hence  ~JT     ^'  ^  =  ^ 

P 

Similarly 

PD       PD 

But  since  b  and  d  are  points  of  equal  potential, 


TT  fl         11  R  P  V          QX 

Hence,  finally,      —  =  — '  or          X  =  — —  • 

JL  (^  •* 

As  soon,  then,  as  a  condition  of  balance  has  been  found,  X  is 
obtainable  at  once  in  terms  of  the  known  resistances  P,  Q,  and  ^. 
In  the  slide-wire  form  of  bridge  (Fig.  45)  abc  is  a  uniform  wire. 
Since  in  this  case  resistance  is  proportional  to  length,  P/Q  is  simply 
the  ratio  of  the  lengths  P  and  Q.  The  heavy  connecting  strips  of 
brass  have  negligible  resistances. 


MEASUREMENT  OF  RESISTANCE 


63 


44.  The  mirror  D' Arson val  galvanometer.  The  accuracy  with 
which  the  point  b  (Fig.  45)  can  be  located,  and  hence  the  accuracy 
with  which  Xcaii  be  found,  evidently  depends  upon  the  sensitiveness 


FIG.  45 

of  the  galvanometer  G.  There  are  two  general  types  of  galvanometer. 
The  first,  like  the  tangent  galvanometer  (Fig.  22,  p.  35),  consists  of 
a  needle  suspended  in  the  middle 
of  a  coil  of  wire  and  hanging 
parallel  to  the  plane  of  the  coil. 
When  a  current  flows  through  the 
coil  the  needle  tends  to  set  itself 
at  right  angles  to  the  plane  of  the 
coil.  This  form  of  galvanometer 
becomes  very  sensitive  when  the 
coil  is  brought  very  close  to  the 
needle  and  the  magnetic  field  in 
which  the  needle  hangs  is  made 
very  weak  (see  sect.  57).  But  a 
weak  field  means  that  the  galva- 
nometer is  very  susceptible  to 
external  magnetic  changes.  For  FIG.  46 

this  reason  this  so-called  Thomson  form  of  galvanometer  is  much 
less  convenient  for  general  laboratory  work  than  is  the  D'Arsonval, 
or  moving-coil,  form.  This  form  (see  Fig.  46,  and  also  Fig.  60,  p.  80) 


64  ELECTRICITY,  SOUND,  AND  LIGHT 

differs  from  the  Thomson  in  that  the  current  is  made  to  pass 
through  a  movable  coil  suspended  in  the  field  of  a  fixed  magnet, 
instead  of  passing  through  a  fixed  coil  at  the  center  of  which  a 
movable  magnet  is  suspended.  The  plane  of  the  coil  is  initially 
parallel  to  the  hues  of  magnetic  force.  As  soon  as  the  current 
traverses  the  coil  it  tends  to  rotate  so  as  to  place  its  plane  at 
right  angles  to  the  direction  of  the  magnetic  lines.  A  mirror  M 
is  attached  to  the  coil  so  that,  with  the  aid  of  a  telescope  and 
scale  placed  at  a  distance,  an  extremely  minute  deflection  of  the 
coil  can  be  detected.  Since  in  this  galvanometer  the  magnetic 
field  in  which  the  coil  is  suspended  is  very  powerful,  slight 
changes  in  the  external  magnetic  conditions  have  no  observable 
influence  upon  the  galvanometer  readings. 

EXPERIMENT  5 

Object.  To  test  as  accurately  as  possible,  by  means  of  Wheatstone's 
bridge,  the  laws  of  series  and  parallel  connections. 

Directions.  I.  Setting  up  the  galvanometer.  After  setting  the  coil  approxi- 
mately parallel  to  the  direction  of  the  magnetic  field,  and  leveling  the  gal- 
vanometer until  there  is  perfect  freedom  in  the  swing  of  the  coil,  shut  out 
all  air  currents  by  covering  the  galvanometer  with  a  box  provided  with  a 
glass  window  in  front.  Then,  setting  the  scale  and  telescope  at  a  distance 
of  about  a  meter  from  the  galvanometer,  focus  the  telescope  upon  the  re- 
flected image  of  the  scale,  as  follows.  Locate  first  the  image  of  your  eye  in 
the  galvanometer  mirror  and  set  the  telescope  pointing  approximately  along 
the  line  of  sight.  Then,  sighting  along  the  telescope,  —  not  through  it,  — 
adjust  the  scale  vertically  until  its  reflected  image  is  seen  in  the  mirror. 
Next  focus  the  eyepiece  of  the  telescope  until  the  cross  hairs  appear  most 
distinct.  This  is  accomplished  by  sliding  the  eyepiece  alone  in  or  out 
of  the  telescope  tube.  Now,  leaving  the  eyepiece  as  adjusted,  focus  the 
telescope  until  the  image  of  the  scale,  as  seen  in  the  telescope,  coincides 
with  the  image  of  the  cross  hairs.  This  coincidence  is  obtained  by  the 
method  of  parallax  as  follows.  The  head  is  moved  slightly  from  side 
to  side  while  the  scale  is  viewed  through  the  eyepiece.  If  there  is  no 
apparent  motion  of  the  cross  hairs  with  respect  to  the  scale,  the  image 
of  the  scale  formed  by  the  objective  is  exactly  coincident  with  the  cross 
hairs,  and  observations  may  be  begun.  If  the  cross  hairs  appear  to  move 
to  the  right  across  the  scale  when  the  head  moves  to  the  right,  the  cross 
hairs  are  farther  from  the  eye  than  is  the  image  of  the  scale,  so  that  the 
draw  tube  .of  the  telescope  should  be  pulled  back  slightly  toward  the  eye. 


MEASUKEMENT   OF  EESISTANCE 


65 


FIG.  47 


If  the  cross  hairs  appear  to  move  over  the  scale  in  a  direction  opposite  to 
that  of  the  motion  of  the  head,  the  draw  tube  should  be  pushed  in  until 
all  apparent  motion  vanishes. 

II.  Connections.    Following  Figure  45  connect  the  unknown  resistance 
xl  to  the  bridge  in  the  place  of  X.    Also  connect  a  known  resistance  at  R. 
Both  of  these  connections  should  be  made  of  heavy  strips  of  copper  of  neg- 
ligible resistance.     The  resistance  R  may  be  either  a  single  standardized 
coil,  or  a  resistance  box.    Figure  47 

shows  the  manner  in  which  the  coils 
in  such  a  box  are  connected.  When 
a  plug  is  taken  out  the  current  is 
obliged  to  pass  through  the  resist- 
ance coil  which  lies  beneath  it  in 
the  box.  When  the  plug  is  put  in 
this  particular  resistance  is  cut  out. 
Kl  and  K2  are  keys  which  may  be 
combined,  if  desired,  into  a  double 
key  of  the  form  shown  in  Figure  48, 

but  in  this  case  the  battery  circuit  should  be  arranged  to  close  through  the 
upper  contact  points,  the  galvanometer  circuit  through  the  lower  contact 
points.  This  is  because  the  equation  for  the  bridge  was  deduced  from 
Ohm's  law,  which  holds  only  for  constant  currents,  not  for  variable  currents 
such  as  exist  at  the  instant  at  which  the  current  of  a  battery  is  closed 
(see  Chap.  XIV).  Kz  is  a  damping  key  which  merely  short  circuits  the 
galvanometer.  It  is  used  for  bringing  the  coil  quickly  to  rest  (see  Chap.  XII). 

Determine  first  the  value  of  xl 
as  explained  in  the  next  para- 
graph. Then  in  a  similar 
manner  determine  x2.  From 
these  values  calculate  the  com- 
bined resistance  of  x1  and  x2, 
first,  when  connected  in  series 
(xs),  and  second,  when  con- 
nected in  parallel  (xp).  Then, 

FIG.  48  •  j        •          •  -v 

using  xl  and  £0  in  series  as  A, 

determine  the  value  by  observation  and  compare  with  the  calculated  value. 
Make  a  similar  determination  and  comparison  for  the  parallel  connection. 

III.  Readings.  If  the  known  resistance  R  is  adjustable,  i.e.  a  resistance 
box,  remove  some  plug,  close  the  short-circuiting  key  Ks,  and,  setting  the 
slider  about  the  middle  of  the  bridge,  press  first  the  battery  key  Kl  and 
then  the  galvanometer  key  K0.    Notice  the  direction  of  the  galvanometer 
deflection.    Now  try  in  the  same  way  some  widely  different  resistance  for  R. 
If  the  deflections  are  in  the  same  direction,  both  of  these  values  of  R  lie  on 
the  same  side  of  the  value  necessary  for  a  balance.    If  the  deflections  are 


66  ELECTRICITY,  SOUND,  AND  LIGHT 

in  opposite  directions,  the  value  sought  lies  between  those  used.  Select 
some  value  of  R  which  will  bring  the  position  of  the  slider  near  the  mid- 
dle of  the  bridge  and  complete  the  adjustment  by  moving  the  slider.  If 
the  known  resistance  is  not  adjustable,  try  two  different  positions  of  the 
slider  to  ge't  an  idea  of  the  necessary  position.  The  short-circuiting  key 
Ks  was  only  closed  at  first  in  order  to  prevent  violent  throws.  As  soon  as 
a  first  approximation  toward  a  balance  has  been  made,  it  should  be  opened 
and  the  accurate  setting  made  with  it  in  this  condition.  After  each  deflec- 
tion the  coil  may  be  brought  quickly  to  rest  by  closing  the  key  7\3. 

In  order  to  find  the  position  of  the  slider  for  which  there  is  no  deflection, 
find  two  points  which  correspond  to  the  smallest  observable  deflections  to 
the  right  and  to  the  left,  K3  being  open.  If  these  points  are  more  than 
.2  or  .3  mm.  apart,  the  galvanometer  is  not  sufficiently  sensitiv3.  The  point 
midway  between  these  two  is  the  point  of  balance.  Since  imperfect  con- 
tacts in  the  resistance  box  may  lead  to  erroneous  results,  it  may  be  well  to 
try  the  point  of  balance  when  the  resistance  R  is  made  up  of  different  com- 
binations of  the  resistance  coils  of  the  box.  If  a  large  variation  results, 
clean  the  plugs  by  wiping  them  with  a  cloth  wet  with  benzine.*  Do  not 
push  down  hard  on  the  plugs ;  merely  twist  them  until  you  are  sure  that  they  make 
good  contacts.  Be  sure  that  all  other  contacts  are  firm. 

In  order  to  minimize  the  error  due  to  lack  of  uniformity  in  the  bridge 
wire  and  also  to  eliminate  thermo-electric  currents,  f  take  the  mean  of  two 
determinations  of  X,  making  the  second  determination  as  follows.  Inter- 
change the  positions  of  R  and  X  and  obtain  values  P'  and  Q'  for  the  lengths 
on  the  bridge  wire. 

EXAMPLE 

The  resistance  R  was  given  a  value  of  20  ohms  and  remained  unchanged 
throughout  the  experiment.  For  xl  the  point  of  balance  on  the  bridge 
wire  was  found  at  P  =  30.41  cm.  and  Q  =  19.59  cm.  When  R  and  xl  were 
interchanged  P  was  found  to  be  30.49,  and  Q,  19.51.  The  average  value 

19  55 

of  xl  was,  therefore,  —  - —  x  20,  or  12.84  ohms.    Similarly  x2  was  found  to 
30.45 

be  21.07  ohms.  Also  #«  was  found  to  be  34.00  ohms,  a  value  which  agreed 
to  within  .3  per  cent  with  the  calculated  value  of  33.91  ohms,  obtained  by 
adding  xl  and  xv  The  value  found  for  xp  was  7.93  ohms,  which  agreed 
well  with  the  calculated  value  of  7.94,  found  from  the  law  for  parallel  con- 
nections by  taking  the  reciprocal  of  the  sum  of  the  reciprocals  of  xl  and  x2. 

*  The  chief  source  of  error  in  all  work  with  resistance  boxes  lies  in  imper- 
fect contacts  at  the  plugs.  Hence  it  is  of  great  importance  to  have  the  plugs 
scrupulously  clean,  and  to  twist  each  plug  in  the  plug  seat  at  the  time  of  its 
insertion. 

t  Thermo-electric  currents  are  discussed  in  Chapters  VI  and  X.  This  inter- 
change of  R  and  X  eliminates  these  currents  only  partially. 


CHAPTEE  VI 

TEMPERATURE   COEFFICIENT  OF  RESISTANCE;    SPECIFIC 
RESISTANCE 

45.  Disadvantages  in  the  slide-wire  form  of  Wheatstone's 
bridge.  For  accurate  work  the  form  of  Wheatstone's  bridge 
described  in  the  preceding  experiment  is  open  to  four  serious 
objections.  First,  the  bridge  wire  may  not  be  uniform  in  diam- 
eter and  resistance  throughout  its  length.  The  exchange  in  posi- 
tion of  R  and  X  tends  to  minimize  the  error  thus  introduced 
in  assuming  P  and  Q  proportional  to  their  lengths;  but  since, 
except  for  a  point  of  balance  at  the  center  of  the  wire,  a  certain 
portion  of  the  wire  is  included  in  both  positions  in  the  larger  of 
the  two  quantities  P  or  Q,  the  error  mentioned  is  not  entirely 
eliminated.  In  general,  however,  the  error  due  to  this  cause  is 
small. 

A  second  and  more  serious  objection  is  this.  Since  the  wire  is 
in  general  not  more  than  a  meter  in  length  and  must,  for  mechan- 
ical reasons,  be  of  considerable  diameter,  its  resistance  is  small. 
The  fall  in  potential  which  takes  place  along  the  wire  is  conse- 
quently small,  so  that  a  small  change  in  the  position  of  the  slide 
causes  but  a  small  variation  in  the  P.D.  between  the  terminals  of 
the  galvanometer  and  consequently  but  a  small  deflection. 

A  third  objection  is  that  thermo-electric  differences  of  poten- 
tial are  introduced  which  cannot  rigorously  be  eliminated  by 
the  interchange  of  R  and  X.  A  thermo-electric  difference  in 
potential  exists  between  the  two  junctions  of  a  metal  with  a 
second  metal  if  these  junctions  are  at  different  temperatures  (see 
Chap.  X).  The  source  of  this  error  in  the  wire  bridge  is  largely 
the  result  of  the  heating  by  the  hand  of  the  junction  of  the  brass 
slider  and  the  German  silver  wire.  Between  this  junction  and  one 
of  the  other  junctions  of  the  wire  with  the  brass  strips  of  the 

67 


68  ELECTRICITY,  SOUND,  AND  LIGHT 

bridge  there  may  exist  a  P.D.,  causing  a  deflection  of  the  galva- 
nometer even  when  Py  Q,  X,  and  R  are  in  balance.  The  point  of 
balance  actually  observed  is  where  this  P.D.  is  counteracted  by  a 
small  P.D.  in  the  opposite  direction,  due  to  having  the  resistances 
slightly  out  of  balance.  Since  the  heating  due  to  the  presence  of 
the  hand  is  not  constant,  the  assumption  of  a  constant  thermal 
current  is  not  rigorous,  and  the  interchange  of  R  and  X  does  not 
entirely  eliminate  the  error.  Another  place  where  this  error  may 
be  introduced  is  at  the  brass-platinum  contact  of  the  key  in  the 
galvanometer  circuit.  Since  the  platinum  contact  is  thin,  and 
consequently  cannot  differ  appreciably  in  temperature  at  its  two 
junctions  with  the  brass  of  the  key,  the  error  here  introduced  is 
small.  On  the  assumption  of  a  constant  error  it  would  be  elimi- 
nated entirely  by  reversing  the  connections  of  the  battery,  since 
this  reverses  the  directions  of  all  the  differences  of  potential  in 
the  bridge  except  those  due  to  the  thermal  effects. 

A  fourth  objection  lies  in  the  fact  that  only  resistances  of 
about  the  same  order  of  magnitude  may  be  compared.  For  evi- 
dently the  observational  error  introduced  in  setting  the  slider  or 
reading  its  position  causes  a  minimum  error  in  the  determination 
of  the  resistance  when  P  and  Q  are  the  same,  i.e.  when  R  and 
X  are  equal.  If  R  and  X  are  widely  different,  this  error  may 
become  quite  large. 

46.  The  post-office-box  form  of  Wheatstone's  bridge.  A  form 
of  bridge  known  from  its  original  use  by  the  telegraph  department 
of  the  British  post  office  as  the  post-office-box  bridge  overcomes 
the  first,  second,  and  fourth  of  these  objections,  and  also  reduces 
largely  the  errors  due  to  thermo-electromotive  forces.  It  possesses 
also  the  advantage  of  compactness,  since  the  resistances  P,  Qt 
and  R,  as  well  as  the  keys  for  the  battery  and  galvanometer 
circuits,  are  contained  in  the  same  box. 

The  essential  difference  between  the  box  and  slide-wire  forms 
consists  in  the  substitution  for  ab  and  lc  (see  Figs.  44  and  45)  of 
two  series  of  accurately  determined  resistance  coils.  This  is  shown 
diagrammatically  in  Figure  49.  The  branches  ab  and  lc  consist 
of  1000-,  100-,  10-,  and  1-  ohm  coils.  The  branch  ad  is  merely  the 
ordinary  set  of  coils  to  be  found  in  a  resistance  box.  The  ratio  of 


MEASUREMENT   OF   SPECIFIC   RESISTANCE 


69 


P  to  Q  is  limited  to  values  of  1000,  100,  10,  1,  T^,  T^,  and 
In  the  use  of  this  form  a  fixed  ratio  is  chosen  for  P/Q,  and  the 
resistance  7?  is  varied  until  a  balance  is  obtained.    For  example, 

7 


w 


100 


WO 


woo 


WOO 


FIG.  49 


in  measuring  a  resistance  X,  known  to  lie  between  3  and  4  ohms, 
the  ratio  P/Q  was  made  1000,  and  E  for  a  balance  was  found  to 
be  3682  ohms,  thus  giving  X=  3.682  ohms. 


/^~N 


EH  ED  d]  IZH  CDiCZi 

; 

CD  EH  a  cn  cula 


FIG.  50 


In  practice  the  boxes  are  not  made  in  the  form  shown  in 
Figure  49,  which  has  been  introduced  as  a  transition  between 
Figure  44  and  Figure  50,  in  which  is  shown  the  ordinary  scheme 


70  ELECTEICITY,  SOUND,  AND  LIGHT 

of  connections.  The  dotted  lines  indicate  connections  made  within 
the  box.  Figures  51  a  and  51  b  show,  in  diagram  and  in  perspective, 
a  box  similar  to  that  represented  diagrammatically  in  Figure  50, 


OOOOGGGO 


,., 

R 


TJA.  GA. 


FIG.  51  a 


the  only  difference  being  that  in  Figure  51  the  battery  is  con- 
nected between  the  points  b  and  d  of  Figure  49,  and  the  gal- 
vanometer between  the  points  a  and  c.  This  has  no  effect  upon 


FIG.  516 


the  equation  of  balance,  for  in  considering  the  diagram  of  the 
Wheatstone  bridge  it  is  evident  that  if  the  battery  and  galva- 
nometer are  interchanged,  the  condition  for  equilibrium  becomes 


MEASUREMENT   OF   SPECIFIC  RESISTANCE         71 

P/B  =  Q/X.  But  this  is  true  if  P/Q  =  li/X.  That  is,  the  gal- 
vanometer and  battery  are  interchangeable  in  position  without 
affecting  the  relation  P/Q  =  R/X* 

47.  Temperature  coefficient  of  resistance.  As  was  stated  on 
page  60,  the  resistance  of  a  conductor  depends  upon  its  tem- 
perature. The  temperature  coefficient  of  resistance  of  a  conductor 
is  denned  as  the  ratio  between  the  change  in  resistance  per  degree 
change  in  temperature  and  the  resistance  at  0°C.  It  will  be  seen 
that  the  definition  is  altogether  analogous  to  that  given  for  the 
coefficient  of  expansion  of  a  gas.  In  symbols,  if  Rt  and  RQ  rep- 
resent the  values  of  the  resistance  of  any  conductor  at  £CC.  and 
0°C.  respectively,  then  the  temperature  coefficient  of  resistance 
a  is  denned  by  the  equation 


This  coefficient  is  positive  for  the  metals,  and  has  a  value  of 
approximately  .0038  for  all  pure  metals.  It  is  negative  for  car- 
bon ;  that  is,  the  hot  resistance  of  an  electric-light  carbon  is  less 
than  its  cold  resistance.  It  will  be  seen  for  the  above  equation 
that  the  resistance  Rt  of  a  conductor  at  any  temperature  may 

be  written  p       p  n        A  /9N 

Mt  —  HQ(L+at).  (2) 

48.  Specific  resistance.  The  specific  resistance  of  a  substance 
at  any  temperature  is  denned  as  the  resistance  between  two  oppo- 
site faces  of  a  centimeter  cube  of  the  substance.  Since  the  resist- 
ance of  a  conductor  varies  directly  as  its  length,  and  inversely  as 

*  The  reason  that  post-office-box  bridges  are  commonly  connected  as  in  Fig- 
ure 51,  rather  than  as  in  Figure  50,  may  be  seen  from  the  following  rule  taken 
from  Maxwell's  "Electricity  and  Magnetism,"  Art.  349.  The  deduction  of 
the  rule  cannot  be  taken  up  in  a  text  of  this  scope.  "  The  rule,  therefore,  for 
obtaining  the  greatest  galvanometer  deflection  in  any  given  system  is  as  follows. 
Of  the  two  resistances,  that  of  the  battery  and  that  of  the  galvanometer,  connect 
the  greater  resistance  so  as  to  join  the  two  greatest  to  the  two  least  of  the  four 
other  resistances."1"1  Since  a  galvanometer  will  practically  always  have  a  higher 
resistance  than  an  ordinary  battery,  and  since,  in  the  use  of  the  post-office 
box  (Fig.  49),  the  resistance  E  is  almost  always  larger  than  X,  and,  therefore, 
P  larger  than  Q,  it  will  be  seen  that,  in  accordance  with  the  rule,  the  galva- 
nometer should  be  connected  across  ac,  and  the  batteiy  across  bd,  if  the  highest 
sensibility  is  to  be  obtained. 


ELECTRICITY,  SOUND,  AND  LIGHT 


its  cross-sectional  area,  the  resistance  R  of  any  conductor,  at  the 
temperature  for  which  its  specific  resistance  r  is  known,  may  be 
written  7 


=  — ,        or 


RA 


(3) 


in  which  /  is  the  length  in  centimeters,  and  A  the  average  area 
of  cross  section  expressed  in  square  centimeters.  Knowing,  then, 
the  resistance  R  at  a  given  temperature  and  the  dimensions  of  a 
conductor,  its  specific  resistance  at  that  temperature  may  be 
readily  calculated  from  (3),  and  if  its  temperature  coefficient  is 
also  known,  its  specific  resistance  at  0°C.  may  then  be  obtained 
from  (2)  by  throwing  it  into  the  form 


(4) 


at 


EXPERIMENT  6 

(A)  Object.    To  determine  the  temperature  coefficient  of  a  commercial 
copper  conductor. 

Directions.    I.  Connections.    The  copper  wire  of  which  it  is  desired  to 
find  the  temperature  coefficient  of  resistance  is  wound  on  a  wooden  frame 

and  supported  by  a  wooden  top 
within  a  thin,  brass  tube  closed 
at  the  lower  end.  Heavy  copper 
wires  pass  from  the  coil  through 
the  cover  (see  Fig.  52).  An  open- 
ing in  the  cover  admits  a  ther- 
mometer supported  so  as  not  to 
touch  the  metal.  This  tube  is 
immersed  in  a  large  vessel  of 
water  to  which  the  heat  is  applied. 
The  resistance  of  the  wire  is 
determined  by  means  of  a  post- 
office  box,  first  at  the  temperature 
of  the  room,  then  at  a  series  of 
higher  temperatures.  This  data 
is  plotted  as  a  curve  on  coo'rdi- 
:  S=:5B'  nate  paper,  using  temperatures 

for   abscissas  and   resistances  as 

ordinates  (see  Fig.  53).    The  curve,  which  is  a  straight  line,  may  be  pro- 
longed backwards,  and  its  intercept  on  the  axis  of  resistances  (i.e.  on  the 


MEASUREMENT  OF   SPECIFIC   RESISTANCE 


73 


ordinate  corresponding  to  the  temperature  0°C.)  taken  as  R0.    The  slope 


of  this  line,  namely 


divided  by  RQ  gives  a. 


II.  Observations.  Measure  the  resistance  of  the  coil,  including  the  con- 
necting wires,  at  the  temperature  of  the  room,  as  follows.     Make  P/Q  j§, 


Temp.  Resist 

21.2°  0.814 

34.0°  0.852 

47.3s  0.893 
0.935 

71.5  3  0.966 
.961  -.751 

70.0 
oc  =0.00402 


FIG.  53 


and  vary  R  until  a  balance  is  obtained  to  within  1  ohm.  (For  example, 
3  ohms  too  small,  4  ohms  too  large.)  It  will  be  best  to  provide  the  gal- 
vanometer with  a  damping  key  as  in  Experiment  5,  and  to  keep  the  latter 
closed  during  this  operation.  Now  make  P/Q  i°A  R  will  be  between  300 


74  ELECTRICITY,  SOUND,  AND  LIGHT 

and  400  ohms.  Again  find  the  value  to  within  1  ohm.  Suppose  R  is  found 
to  lie  between  368  and  369.  Then  make  P/Q  ^?°-  and  find  Pi  to  the  nearest 
ohm.  If  R  lies  between  3682  and  3683,  take  the  number  which  gave  the 
smallest  deflection,  e.g.  3683.  Then  X  =  3.683.  If  the  galvanometer  is 
sensitive,  the  accuracy  can  be  pushed  at  least  one  place  farther  by  the  method 
given  below  in  (B),  but  with  the  accuracy  attainable  in  the  temperature 
readings  of  this  experiment,  it  is  scarcely  advisable  to  attempt  greater 
refinements  in  the  resistance  measurements. 

Keeping  P/Q  =  -1(\00,  raise  the  temperature  of  the  water  about  10°  above 
that  of  the  room.  Stir  thoroughly,  and  by  regulating  the  flame  of  the  Bunsen 
burner  keep  the  temperature  of  the  water  as  constant  as  possible.  Since 
the  air  surrounding  the  coil  is  a  poor  conductor  of  heat,  the  coil  will  be 
slow  in  rising  to  the  temperature  of  the  water.  Follow  its  rise  by  varying 
the  resistance  in  the  post-office  box  and  take  a  reading  of  this  resistance  as 
soon  as  it  becomes  constant.  Note  the  temperature  of  the  thermometer  in 
the  inner  tube. 

Next  begin  to  plot  your  readings  as  follows.  Plot  temperatures  along 
the  X  axis  of  the  coordinate  sheet,  using,  for  example,  1  division  to  repre- 
sent 2°C.,  or  choosing  some  other  scale  which  will  make  70  or  80  degrees 
occupy  nearly  the  full  width  of  the  page.  Plot  resistances  along  the  Y  axis, 
using  any  convenient  scale.  The  method  of  selecting  a  scale  which  will 
make  the  fullest  use  of  the  sheet  may  be  seen  from  the  example  presented 
in  Figure  53.  The  first  two  observed  temperatures  were  21.2°C.  and  34.0°C. 
The  increase  in  resistance  corresponding  to  this  increase  of  12.8°  in  tem- 
perature was  .038  ohm.  In  raising  the  temperature  from  0°C.  to  70°C.  it 
was  estimated  that  there  should  be  an  increase  of  about  5.5  times  this 
amount,  or  approximately  .210  ohm.  Now  there  were  on  the  particular 
sheet  shown  in  the  figure  110  available  spaces  along  the  Y  axis.  Dividing 
the  approximate  number  of  thousandths  of  an  ohm,  namely  210,  through 
which  the  resistance  was  expected  to  be  raised,  by  the  number  of  available 
divisions,  gave  1.95  as  the  number  of  thousandths  of  an  ohm  to  be  repre- 
sented by  1  space,  if  the  sheet  was  to  be  completely  utilized.  Since  this 
number  was  very  inconvenient  both  for  plotting  the  graph  and  for  reading 
the  values  from  it,  the  nearest  whole  number  above  1.95,  namely  2,  was 
taken  as  the  number  of  thousandths  of  an  ohm  to  be  represented  by  1 
division.  The  general  rule  for  this  choice  would  be  as  follows.  Select  the 
nearest  whole  number  larger  than  the  quotient  found  as  above  except  where  this 
number  would  be  3,  6,  7,  or  9,  in  which  case  it  would  be  more  convenient  to  choose 
4,  5,  8,  or  10  respectively. 

Do  not  call  the  intersection  of  the  two  axes  zero  resistance,  but  make 
it  instead  some  convenient  number  which  is  about  10  per  cent  less  than 
the  resistance  of  the  coil  at  the  temperature  of  the  room  (see  Fig.  53). 

Having  plotted  the  first  two  points,  raise  the  temperature  by  about  ten- 
degree  steps,  and  plot  each  point  as  soon  as  it  is  obtained.  If  any  point 


MEASUREMENT  OF  SPECIFIC   RESISTANCE 


75 


does  not  lie  on,  or  very  near  to,  the  right  line  joining  the  first  two  points, 
repeat  the  observation  before  raising  the  temperature  farther.  If  results 
continue  to  be  erratic,  wipe  all  the  plugs  with  a  clean  cloth  moistened 
with  benzine  and  look  to  your  contacts.  For  example,  turn  each  plug 
until  it  catches.  Do  not,  however,  push  down  hard  upon  the  plugs, 
as  this  may  spring  the  lugs  and  thus  injure  the  box.  After  carrying 
the  temperature  to  70°C.  or  80°C.,  disconnect  the  connecting  wires  from 
the  heater,  connect  their  terminals  to  each  other,  and  measure  the  resist- 
ance of  these  wires. 

III.  Calculation*.  Draw  a  straight  line  which  will  come  as  near  as  pos- 
sible to  all  of  the  plotted  points.  Produce  the  curve  thus  found  until  it 
cuts  the  Y  axis  (the  line  of  zero  temperature) ,  and  take  this  point  of  inter- 
section as  the  resistance  of  the  coil  at  0°C.  Subtract  from  this  value  the 
resistance  of  the  connecting  wires  to  find  RQ.  Find  a  by  dividing  the  slope 
of  the  curve  by  RQ  (see  Fig.  5o) . 

(B)  Object.  To  find  the  specific  resistance  of  copper  at  0°C. 

Directions.  In  order  to  determine  a  specific  resistance  the  dimensions  of 
the  wire  must  be  accurately  measured.    This  condition  necessitates  a  large 
wire,  and  hence  one  of  small  resistance.     But  neither  the  slide-wire  nor 
the  post-office  form  of  bridge  is  suitable, 
without  modification,  for  the  measurement 
of  a  resistance  which  is  so  small  as  to  be 
comparable  with  the  resistances  of  the  con- 
tacts or  of  the  connecting   strips.     These 
are  often  several  thousandths  of   an  ohm. 
A  very  satisfactory  method  of  eliminating 
all  errors  due  to  contacts,  connecting  wires, 
or    thermal    electromotive    forces,    and    of 
measuring  with  considerable  accuracy  a  re- 
sistance of  a  few  thousandths  of  an  ohm  is 
as  follows. 

Let  large  connecting  wires  a  and  b  from 
the  post-office  box  terminate  in  two  holes 

1  and  3  (Fig.  54)  bored  in  a  wooden  block  and  filled  with  mercury. 
Let  an  auxiliary  wire  c  connect  holes  1  and  £,  and  the  wire  of  unknown 
resistance  X  complete  the  circuit  by  connecting  holes  2  and  3.  Let  the 
surfaces  of  contact  of  all  the  wires  with  the  mercury  be  well  amalgamated 
by  dipping  the  ends  of  the  wires  into  nitric  acid,  then  into  mercury,  then 
rubbing  dry  with  filter  paper.  Let  P/Q  be  made  J-0^"  and  let  R  be  varied 
until  a  change  of  1  ohm  causes  a  change  in  the  direction  of  the  deflection. 
Then  take  the  permanent  deflection  produced  by  closing  Kz  (Fig.  45)  when 
R  has  the  lower  value,  say  6  ohms.  Then  add  1  ohm  to  R  and  take  the 
permanent  deflection  in  the  other  direction.  If,  for  example,  the  first  de- 
flection were  4.35  mm.  and  the  second  62.70  mm.,  then  the  value  of  R  for 


FIG.  54 


o 


76  ELECTRICITY,  SOUND,  AND  LIGHT 

a  perfect  balance  would  have  been  0  +  -  —  =  G.065  ohms.     Now. 

4.35  +  62.70 

with  as  little  delay  as  possible,  remove  J\T,  transfer  b  from  3  to  2  (see  Fig.  55) , 
and  again  find  precisely  as  before  the  value  of  R  which  would  produce  a 
balance.  If  this  is  found  to  be,  for  example,  4.107  ohms,  then  obviously 

X  —  —  —  =  .001958  ohm.     Repeat  the  observations  and  see  how 

nearly  the  difference  between  the  two  values  of  R  can  be  checked.  Changes 
may  occur  in  the  individual  values  of  R  because  of  temperature  changes, 

but  the  differences  should  remain  very 
nearly  constant.  Next  take  the  mean  of 
some  ten  or  more  measurements  of  the 
diameter  of  the  wire,  using  the  micrometer 
caliper.  Measure  with  a  tape  the  distance 

/between   the    two    points    on    the  wire    to 
/  "  which   it  was   immersed   in  the   mercury. 

pIG    55  From    these    data    and    the    temperature 

coefficient    as    found    in  (A)   compute    the 

specific  resistance  of  commercial  copper  at  0°C.  by  the  aid  of  equa- 
tions (3)  and  (4). 

EXAMPLE 

(A)  The    curve   plotted   as    above    (Fig.  53)    was    the    record    of   this 
experiment.     Tabulated   in  one  corner   are  found  the  various  values  of 
the  resistance  and  temperature,  the  value  of  7t0,  the  value  of  the  slope 

of  the  line  — —  >   and  the  temperature  coefficient  ex.. 

(B)  Total  resistance  =  .00595  ohm  ;  temperature  =  26°  C. ;  resistance  of 
connecting  wires  and  contacts  =  .00403  ohm;  length  of  wire  =  137 cm.; 
mean  radius  =  .205cm.;  therefore  specific  resistance  at  0°C.  =.00000107 
ohm  =  1.G7  X  10~6  ohms  =  1670  absolute  units. 


CHAPTER  VII 

GALVANOMETER  CONSTANT  OF  A  MOVING-COIL 
GALVANOMETER 

49.  Galvanometer  constant.    In  the    formula  for  the  current 
flowing  through  a  tangent  galvanometer,  namely 

ffrttmti 

•/=-2^T' 

it  is  seen  that  in  general  the  current  is  not  proportional  to  the 
deflection,  although  it  would  be  so  if  deflections  were  always  kept 
so  small  that  tan  6  were  approximately  equal  to  6.  In  galvanom- 
eters in  which  the  current  is  proportional  to  the  deflection,  the 
reduction  factor  by  which  the  angle  of  deflection  must  be  multi- 
plied in  order  to  give  the  current  is  known  as  the  galvanometer 
constant.  Thus  if  /represents  the  current  which  produces  in  such 
a  galvanometer  a  deflection  of  0  radians,  then  the  definition  of  the 
galvanometer  constant  K  is  given  by  the  equation 

J=M       or       A-=|-  (1) 

That  is,  the  galvanometer  constant  is  defined  as  the  constant  ratio 
between  the  current  and  the  deflection  produced  "by  it.  This  ratio 
for  small  angles  is  still  called  in  many  cases  the  galvanometer 
constant,  even  though  the  instrument  be  one  with  which  the  ratio 
may  not  remain  constant  for  large  angles.  It  is  the  object  of  this 
discussion  to  find  the  expression  for  this  constant  in  the  case  of 
a  moving-coil  (i.e.  a  D'Arsonval)  galvanometer.  To  do  this  it  will 
be  necessary  first  to  reconsider  the  definition  of  unit  current  given 
in  section  27,  page  34. 

50.  Restatement  of   definition  of  unit  current.    Unit  current 
has  been  defined  as  a  current  which,  when  flowing  in  a  conductor 
of  unit  length  bent  into  the  arc  of  a  circle  of  unit  radius,  exerts 
unit  force  on  a  unit   magnetic   pole   lying  in  the  plane  of  the 

77 


78 


ELECTRICITY,  SOUND,  AND  LIGHT 


1    c  m 


\ 


conductor  and  at  the  center  of  the  arc.  Thus  the  unit  pole  at  0 
(Fig.  56)  is  urged  from  the  plane  of  the  paper  toward  the  reader 
with  a  force  of  1  dyne  if  the  current  in  the  wire  has  unit  strength. 

Since  action  and  reaction  are  equal  and 
opposite  in  direction,  the  conductor  is 
urged  from  the  reader  toward  the  paper 
with  an  equal  force.  Now  the  magnetic 
field  due  to  the  isolated  unit  pole  has 
a  strength  at  all  points  on  the  wire  of 
unity,  and  a  direction  at  right  angles  to 
the  wire,  since  its  lines  of  force  are  straight 
lines  radiating  from  it  in  all  directions. 
Instead,  then,  of  stating  the  definition  of 
unit  current  in  terms  of  the  force  which  the  current  exerts  on  the 
magnet,  we  may  state  it  in  terms  of  the  force  which  the  magnetic  field 
exerts  on  the  conductor.  Thus  unit  current  is  that  current  unit  length 
of  which,  when  flowing  in  unit  field  at  right  angles  to  the  direction  of 
the  field,  experiences  unit  force.  In  symbols,  the  force  F,  in  dynes,  ex- 
erted on  /  cm.  of  length  of  a  conductor  carrying  /  electro-magnetic 
units  of  current,  by  a  field  of  strength  &C,  at  right  angles  to  I  is,  by 


o 

FIG.  56 


definition  of  current, 


(2) 


Xieft  Hand 


Magnetic 


51.  The  motor  rule.  The  relation  between  the  directions  of  the 
magnetic  field,  the  current  in  the  conductor,  and  the  force  exerted 
on  the  conductor  may  be  seen 
from  the  figure  to  be  stated  in 
the  following  rule,  known,  from 
its  especial  application  to  the 
direct-current  motor,  as  the 
"  motor  rule."  Extend  the  thumb, 
forefinger,  and  second  finger  of 
the  left  hand  in  directions  at 
right  angles  to  one  another ;  let 
the  forefinger  point  in  the  direc- 
tion of  the  magnetic  field,  and  the 
second  finger  in  the  direction  of  the  current;  the  thuml)  will  then 
point  in  the  direction  of  the  force  acting  on  the  conductor  (Fig.  57). 


FIG.  57 


MOVING-COIL   GALVANOMETER 


79 


52.  Faraday's  explanation  of  electro-magnetic  forces.  The  fact 
that  the  force  between  a  magnet  and  a  current  does  not  act  along 
the  line  connecting  the  magnet  and  the  conductor,  but  at  right 
angles  to  this  line,  constitutes  a  striking  difference  between  electro- 
magnetic forces  and  the  forces  met  with  in  the  study  of  mechan- 
ics ;  for  these  uniformly  act  along  the  lines  connecting  the  acting 
bodies.  Further,  the  forces  of  mechanics,  excepting  gravitation, 
act  only  as  the  result  of  the  transmission  of  stresses  through 
matter.  Because  of  the  difficulty  of  the  conception  of  action  at  a 
distance,  that  is,  action  assumed  to  take  place  without  the  inter- 
vention of  a  medium,  and  in  order  to  admit  of  a  more  perfect  visu- 
alization of  the  actions  of  magnets  and  electrical  charges,  Faraday 
conceived  of  these  electrical  and  magnetic  actions  as  having  their 
seat  in  the  lines  of  force ;  that  is,  he  imagined,  merely  as  an  aid  to 


thinking,  that  these  lines  of  force  constituted  a  sort  of  system  of 
invisible,  stretched,  elastic  bands  endowed  writh  the  following  prop- 
erties: (1)  a  tension  in  the  direction  of  their  length,  and  (2)  a  repul- 
sion at  right  angles  to  this  direction,  so  as  to  cause  them  to  act  on 
one  another  as  if  a  hydrostatic  pressure  existed  at  right  angles  to 
their  direction.  Thus  in  Figure  58,  a,  which  shows  the  field  existing 
between  two  opposite  magnetic  poles,  the  lines  at  the  top  and  bot- 
tom were  thought  of  as  forced  out  by  a  repulsion,  not  balanced,  as 
at  the  center,  by  an  equal  and  opposite  repulsion  due  to  other  lines. 
Again  in  Figure  58,  b,  is  shown  the  field  surrounding  a  wire 
carrying  a  current  toward  the  paper,  while  in  Figure  58,  c,  is  the 
resultant  field  due  to  the  presence  in  the  uniform  field  of  a  of  the 
conductor  of  b.  The  resultant  field  at  p  is  reenforced,  while  that 
at  q  is  weakened.  The  lines  are  therefore  crowded  together  more 
closely  at  p  and  less  so  at  q.  There  results  from  these  hypothetical 


80 


ELECTRICITY,  SOUND,  AND  LIGHT 


characteristics  of  the  lines  an  unbalanced 
repulsion  forcing  the  conductor  across  the 
field  toward  the  bottom  of  the  page,  in 
accordance  with  the  demands  of  the  motor 
rule.  This  mechanical  picture  of  the  action 
of  electric  and  magnetic  forces  has  so  per- 
meated the  literature  of  electricity  and 
magnetism  that  a  familiarity  with  it  is  of 
importance  to  the  student  of  the  subject. 
53.  The  equation  of  the  moving-coil 
galvanometer.  In  Figure  59,  a,  is  shown 
a  diagram  representing  the  magnets  and 
one  single  loop  of  the  coil  of  a  moving-coil 
galvanometer  (see  also  Fig.  60).  Suppose 
that  a  current  of  /units  is  flowing  through 
the  coil  and  that  the  strength  of  the  mag- 
netic field  between  N  and  S  is  3{  units. 

Then  the  left  side  of  the  loop  is  urged  toward  the  reader  with  a 

force  of  JOT  dynes  (sects.  50  and  51)  and  the  right  side  is  urged  away 

from  the  reader  with  an  equal  force. 

Upon  the  horizontal  wires  no  force 

acts,  for  they  carry  currents  parallel 

to  the  direction  of  the  field.   If  the 

distance  between  the  two  sides  is  d 

centimeters,  the  moment  of  force  Fh 

acting  to  twist  the  loop  about  a  ver- 
tical axis  midway  between  its  sides 

is  given  by 


FIG.  59 


Fh  =  II 


(3) 


Since  Id  is  simply  the  area  a  of  the 
loop,  this  equation  may  be  written 


Fh  =  la&C. 


(4) 


If   there   are   n  loops   of    average 
area  a,  the  total  twisting  moment 


FIG.  60 


MOVING-COIL  GALVANOMETER  81 

is  of  course  lancH*',  and  if  WQ  call  an  the  total  area  of  the  coil  and 
represent  it  by  A,  we  have 

~Fh  =  IA&C.  (5) 

This  is  the  value  of  the  couple  which  acts  on  the  coil  so  long  as 
the  plane  of  its  loops  is  parallel  to  the  direction  of  the  field  cK. 
Under  the  influence  of  this  moment  of  force  it  rotates  until  it  is 
brought  to  rest  by  the  restoring  moment  due  to  the  torsion  of  the 
suspending  fiber.  Suppose  that  when  equilibrium  is  established 
the  coil  has  rotated  through  an  angle  0  (see  Fig.  59,  b).  The  couple 
arm  will  then  have  changed  from  d  to  d  cos  0,  so  that  the  moment 
of  force  producing  the  deflection  is  now  IA&C  cos  0  instead  of  IA&C, 
while  the  restoring  moment  is  TJ9,  if  we  represent  by  T0  the  mo- 
ment of  torsion  of  the  suspending  fiber.  Hence  the  equation  of 
equilibrium  of  the  moving-coil  galvanometer  is 

In  deducing  this  expression  we  have  assumed 
a  rectangular  coil,  but  it  can  very  readily  be 
seen  that  the  result  is  precisely  the  same 
whatever  its  shape.  For  any  irregular  coil 
may  be  considered  to  be  made  up  of  infini- 
tesimal rectangular  elements  (see  Fig.  61)  and 
we  have  just  seen  above  that  the  moment  of 
force  acting  on  a  rectangular  coil  is  IctcK  cos0, 
where  a  is  the  area  of  the  rectangular  element. 
The  total  moment  is  therefore  2/#<5f  cos0  or 


cos  0,  where  the  total  area  of  the  coil,  found  by  summing 
up  the  infinitesimal  areas,  is  A. 

Equation  (6)  shows  that  the  current  /  is  not  proportional  to  the 

0 

deflection  6,  but  rather  to  -  -•    If,  however,  0  is  so  small  that 

cose' 

cos  0   may,  without   appreciable   error,   be   taken   as   unity,  then 
equation  (6)  becomes  ™ 


and  it  is  clear  that  the  galvanometer  constant  K,  i.e.  1/6  (see 
eq.  (1)),  is  given  by  _ 


82 


ELECTRICITY,  SOUND,  AND  LIGHT 


This  expression  makes  it  obvious  why  K  is  called  the  galvanometer 
constant,  for  it  involves  only  the  nature  of  the  suspending  fiber  ( TQ), 
the  area  of  the  coil  (A),  and  the  strength  of  the  magnetic  field  (cf(). 
Keeping  clearly  in  mind  the  restriction  limiting  the  determina- 
tion and  use  of  K  to  small  angles,  it  is  possible  to  find  K  for  any 
galvanometer  by  measuring  the  current  which  produces  a  small 
angular  displacement  and  dividing  the  current  by  the  displacement. 
For  some  purposes  it  is  more  convenient  to  have  this  reduction 
factor  expressed,  not  in  terms  of  current  per  radian,  but  in  terms 
of  the  current  necessary  to  produce  a  deflection  of  1  mm.  on  a 
scale  at  a  distance  of  1  in.  The  current  in  amperes  necessary 
to  do  this  is  known  as  the  figure  of  merit  of  the  galvanometer. 
Since  the  mirror  of  a  galvanometer  actually  turns  through  one  half 
the  angle  through  which  the  reflected  ray  is  rotated,  and  since 
1  mm.  is  .001  m.,  it  is  evident  that  the  figure  of  merit  k  of  a  gal- 
vanometer is  7T/2000. 

Again,  instrument  makers  often  express  the  "  sensibility  "  of  a 
galvanometer  in  terms  of  the  number  of  megohms  (million  ohms) 
which  would  have  to  be  placed  in  series  with  it  in  order  to  reduce 
the  deflection  to  1  mm.  on  a  scale  1  m.  distant,  when  a  P.D.  of 

1  volt  exists  between  the  ends  of  this 
resistance.  This  is  obviously  the  re- 
ciprocal of  the  figure  of  merit.  Thus 
a  galvanometer  which  has  a  constant 
K  equal  to  .000004,7  being  measured 
in  amperes,  has  a  figure  of  merit  of 
.000000002  =  2  x!0~9,  and  a  sensi- 
bility of  1/2x10°  =  500  megohms. 
54.  Method  of  determining  K. 
The  scheme  of  connections  which  is 
used  for  determining  K  is  shown 
in  Figure  62.  The  current  from  a 
Daniell  cell  B  is  made  small  by 

inserting    into    the    circuit   a  large 

FIG.  62        ^^  n  .       ..e 

resistance  Rr    Only  a  small  fraction 

of  this  current  is  passed  through  the  galvanometer  G,  which  is 
placed  in  one  arm  of  a  divided  circuit  made  up  on  the  one  side 


MOVING-COIL  GALVANOMETER  83 

of  the  small  resistance  R2  and  on  the  other  side  of  the  large 
resistance  Rs  plus  the  galvanometer  resistance  G.  The  deflection 
corresponding  to  a  given  arrangement  of  the  resistances  is  observed 
and  the  current  passing  through  the  galvanometer  is  calculated 
from  the  reading  (P.D.)  of  the  voltmeter  and  the  values  of  the 
various  resistances.  K  is  then  found  by  dividing  the  current  by 
the  deflection,  the  latter  being  expressed  in  radians. 

The  calculation  of  the  current  follows  directly  from  Ohm's  law. 
If,  as  is  usually  the  case,  the  resistance  of  the  branched  circuit 
from  a  to  b  is  negligible  in  comparison  with  Rv  then  the  current 
II  in  the  main  circuit  is  given  by 


Since  the  potential  difference  between  a  and  b  is  the  same  for 
both  branches,  the  currents,  /  through  the  galvanometer,  and  /2 
through  E2,  will  vary  inversely  as  the  resistances  of  their  respective 
branches.  That  is 

(9) 

Since  the  sum  of  these  two  currents  must  be  the  current  in  the 
main  circuit,  we  have 

/+/,  =  /,.  (10) 

From  (9)  and  (10)  we  obtain 


This  shows  that  in  general  the  current  which  flows  through  one 
side  of  a  shunt  is  obtained  by  multiplying  the  total  current  by  the 
resistance  of  the  other  side  divided  by  the  resistances  of  both  sides. 

From  (8)  and  (11)  w^e  at  once  obtain  I  in  terms  of  the  various 
resistances  and  the  voltmeter  reading  P.D. 

If  this  current  causes  a  deflection  of  d  centimeters  on  a  scale 
D  centimeters  from  the  mirror,  the  angle  through  which  the 
mirror  rotates  (one  half  that  through  which  the  reflected  beam 
rotates)  is  d/1  D.  Hence 

211) 


A  \  (       G 

\       \ 


84  ELECTEICITY,  SOUND,  AND  LIGHT 

55.  Resistance  of  the  galvanometer.    The  resistance  of  the  gal- 
vanometer can  be  found  most  accurately  by  connecting  it  as  the 
unknown  to  a  post-office  box  and  measuring  it  by  the  Wheatstone- 
bridge  method,  using  another  galvanometer.    But  if  J?3  is  large 

(compared  with  G),  G  may  be  found 
very  much  more  conveniently  and 
with  sufficient  accuracy  by  connect- 
ing a  Daniell  cell  B,  a  voltmeter 
V,  a  milliammeter  A,  and  the  galva- 
nometer G  as  in  Figure  63.  The  read- 
ing of  V  in  volts,  divided  by  that 
of  A  in  amperes,  gives  the  resistance 
in  ohms  of  G  and  A  together,  and 
the  resistance  of  A  will,  in  general, 
be  either  negligible  or  kno\vn. 

56.  Direct-reading   moving-coil   galvanometers.    If   the   term 
A3i cos0  in  the  expression  /=  T00/A&C cos#  (see  eq.  (6),  p.  81) 
can  be  made  constant,  the  deflection  will  be  proportional  to  the 
current  for  all  values  of  6.    This  condition  means  that  the  pole 
pieces  must  be  so  shaped  that  the  field  strength  ctf  acting  on  the 
vertical  wires  of  the  coil  in  any  position  shall  be  inversely  pro- 
portional to  the  cosine  of  the  angle  through  which  it  has  been 
deflected.    Instruments  in  which  this  condition  has  been  met  may 
be  used  for  the  measurement  of  widely  different  currents  on  a  scale 
of  uniform  divisions.     The    common,   commercial,   direct-reading 
ammeter  is  an  instrument  of  this  sort. 

The  plan  of  an  ammeter  is  shown  in  Figure  64.  M  is  a  per- 
manent magnet  of  which  aaf  and  W  are  the  shaped  pole  pieces. 
E  is  a  soft-iron  cylindrical  core  which  concentrates  the  field. 
The  coil  c  is  wound  on  a  rectangular  aluminum  frame,  and 
turns  on  pivots  in  jeweled  bearings  against  the  torsion  of  two 
flat  spiral  springs  not  shown  in  the  figure.  The  pointer  D  is 
attached  to  the  coil  and  moves  over  a  scale  S  empirically  cali- 
brated, but  having  practically  constant  divisions.  The  terminals 
of  the  coil  are  attached  to  heavy  wires  wwr  which  connect  with 
the  binding  posts  BB'.  Across  these  wires,  in  shunt  with  the 
coil,  are  a  number  of  small  resistances  r  which  carry  the  greater 


MOVING-COIL  GALVANOMETER 


85 


part  of  the  current,  allowing  only  a  fraction  to  pass  through  the 
movable  coil. 

The  current  flows  through  the  coil  of  the  ammeter  in  a  direc- 
tion such  that  it  produces  the  magnetic  field  indicated  by  the 
small  letters  n  and  s.  The  coil  therefore  tends  to  rotate  to  the 
right.  The  angle  9  is  reckoned  from  a  position  of  the  coil  parallel 
to  the  lines  of  force  NS.  Thus  6  has  a  value  of  about  45°  for  the 
initial  position  of  the  coil  shown  in  the  figure.  Owing  to  its  rotation 


FIG.  64 


o 


from  this  position,  0  decreases  to  zero  when  the  coil  is  parallel  to 
NS,  and  then  increases  to  about  45°  for  a  position  parallel  to  a'b. 
This  means  that  cos  0  increases  to  unity  and  then  decreases.  In 
order  that  3i  cos  0  shall  be  constant  3i  must  be  larger  at  the  tips 
of  the  pole  pieces  a,  a',  b,  and  bf  than  at  the  center.  This  is  accom- 
plished by  giving  the  pole  pieces  the  shape  shown  in  the  figure. 

A  current-measuring  instrument  of  high  resistance  may  be  cali- 
brated to  read  units  of  P.D.,  e.g.  volts,  as  was  explained  on  page  52. 
Such  an  instrument,  of  the  movable-coil  type,  is  the  voltmeter. 


86 


ELECTRICITY,  SOUND,  AND  LIGHT 


FIG.  65  a 


It  differs  essentially  from  the  ammeter  of  Figure  64  only  in  the 
matter  of  resistance.  Instead  of  the  small  resistances  r  in  shunt 
between  the  binding  posts  BB'  there  is  a  single  large  resistance 

in  series  with  the  movable  coil  c. 
57.  Types  of  current-measur- 
ing instruments.  Thus  far  two 
distinct  types  of  instruments  for 
measuring  currents  by  the  mutual 
actions  of  two  magnetic  fields 
have  been  described.  In  the  first 
the  current  is  measured  by  the 
deflection  of  a  magnet  which  as- 
sumes a  position  of  equilibrium 
under  the  action  of  an  external 

field  and  the  field  due  to  a  fixed  coil  through  which  the  cur- 
rent to  be  measured  is  flowing.  The  tangent  galvanometer  is  an 
instrument  of  this  type,  the  external  field  being  that  of  the  earth. 

The  Thomson  galvanometer  described  in 
connection  with  Figure  46  is  also  an  instru- 
ment of  this  type,  but  in  it  the  external 
field  is  in  general  the  resultant  field  due  to 
the  earth  and  a  bar  magnet  attached  to 
the  galvanometer  in  the  manner  shown  in 
Figure  65  a.  This  magnet  is  used  to  oppose 
the  earth's  field  at  the  galvanometer,  and, 
by  weakening  the  resultant  field,  to  increase 
the  sensitiveness  of  the  instrument.  In  the 
most  sensitive  instruments  of  the  Thomson 
type  the  effect  of  the  earth's  field  is  reduced 
practically  to  zero  by  the  use  of  a  so-called 
astatic  system.  This  consists  of  two  oppo- 
sitely directed  sets  of  small  magnetic  needles 
ns  (Fig.  65  a)  mounted  in  the  same  plane 
on  mica  disks  which  are  attached  to  a 
light  rigid  frame  d.  The  earth's  field  obviously  tends  to  make 
one  set  rotate  in  one  direction  and  the  other  in  the  opposite 
direction,  so  that,  if  the  needles  were  exactly  alike,  the  resultant 


FIG.  656 


MOVING-COIL  GALVANOMETER 


87. 


effect  would  be  zero.  In  order  to  avoid  having  the  effect  of  the 
current  011  one  set  opposite  to  its  effect  on  the  other  set,  the  two 
sets  are  hung  in  separate  coils  D  and  E,  through  which  the  current 
flows  in  opposite  directions  as  shown  in  the  figure.  A  commercial 
form  of  this  instrument  is  shown  in  Figure  65  b. 

The  D'Arsonval  galvanometer  and  its  practical  form,  the 
ammeter,  are  the  chief  representatives  of  the  second  type  of 
instruments,  in  which  a  movable  coil  rotates  in  a  fixed  magnetic 
field  due  to  permanent  magnets.  While  instruments  of  this  type 
cannot  be  made  nearly  as  sensitive  as  the 
Thomson  galvanometer,  they  are  much  more 
satisfactory  for  ordinary  work  because  of 
the  fact  that  they  are  not  influenced  by  the 
magnetic  disturbances  due  to  the  electric 
currents  which  flow  within  or  near  to  most 
buildings. 

There  is  a  third  form  of  current-measuring 
instrument  in  which  the  movable  coil  is 
placed  in  the  magnetic  field  due  to  a  fixed 
coil.  The  principle  underlying  many  dif- 
ferent varieties  of  instruments  of  this  type 
may  be  seen  from  Figure  66  a.  A  fixed  coil 
AS  carrying  the  current  to  be  measured 
sets  up  a  magnetic  field  in  which  there  is 
suspended  by  a  helical  spring  T  a  movable 
coil  CD  connected  in  series  to  the  fixed 
coil  and  also  carrying  the  current  to  be 
measured.  Since  the  magnetic  field  due  to 
the  fixed  coil  is  proportional  to  the  current, 

and  since  the  reaction  between  the  current  flowing  in  the  mov- 
able coil  and  this  magnetic  field  is  proportional  to  this  current 
times  the  strength  of  field,  it  follows  that  the  moment  of  force 
tending  to  rotate  the  movable  coil  is  strictly  proportional  to 
the  square  of  the  current,  provided  that  the  relative'  positions 
of  the  coils  are  kept  unaltered.  This  condition  is  satisfied  by 
twisting  the  helical  spring  until  the  torque  it  exerts  balances 
the  moment  acting  on  the  coil.  The  angle  0  through  which  the 


FIG.  60  a 


ELECTRICITY,  SOUND,  AND  LIGHT 


spring  is  twisted  is  proportional  to  this  moment  and  consequently 
to  the  square  of  the  current  I.  That  is, 

IxV0,         or         I=kV0.  (13) 

The  reduction  factor  k,  by  which  the  square  root  of  the  angle 
must  be  multiplied  to  give  the  value  of  the  current,  may  be 
found  from  an  observation  of  the  current  and  the  corresponding 
deflections.  Thereafter  the  current  may  be  found  from  the  relation 

of  equation  (13).  A  commercial 
form  of  this  "dynamometer" 
is  shown  in  Figure  6  6  fr. 

The  great  majority  of  alter- 
nating-current ammeters  and 
voltmeters  are  instruments  of 
the  dynamometer  type.  For,  in 
view  of  the  fact  that  any  change 
in  the  direction  of  the  current 
takes  place  simultaneously  in. 
both  coils,  the  direction  of  the 
torque  remains  always  the  same ; 
so  that,  if  the  alternations  are 
sufficiently  rapid,  an  alternat- 
ing as  well  as  a  direct  current 
will  produce  a  constant  deflec- 

„      pp ,  tion   which    is    proportional  to 

the  mean  square  of  the  current 

strength.  The  dynamometer  also  has  the  advantage  of  having  no 
permanent  magnets  whose  magnetism  may  undergo  slow  changes 
with  time.  This  is  indeed  the  chief  source  of  inaccuracy  in  cali- 
brated instruments  of  the  D'Arsonval  type. 


EXPERIMENT   7 

Object.    T°  find  the  constant  K  of  a  moving-coil  galvanometer. 
Directions.    Find  the  resistance  of  the  galvanometer*  by  either  of  the 
two    methods    stated  above,  as  directed  by  the  instructor.    Connect   the 


*The  galvanometer  should  be  the  ballistic  one  which  it  is  intended  to  use  in 
Experiment  8. 


MOVING-COIL  GALVANOMETER  89 

resistances,  a  voltmeter  and  a  Daniell  cell,  as  in  Figure  62.  See  that  the 
scale  is  normal  to  the  line  drawn  from  the  zero  reading  to  the  mirror. 
Make  7^  =  10,000  ohms  and  Rs  =  1000  ohms.  Make  R2  successively  1,  2, 
and  3  ohms,  and  observe  the  corresponding  deflections.  If  these  values  of 
Rv  J?2,  and  R3  do  not  cause  deflections  of  from  3  to  10  cm.  on  a  scale 
at  1  meter's  distance,  choose  other  values  which  will  keep  the  deflections 
within  about  these  limits.  Since  the  chief  source  of  error  in  this  experi- 
ment lies  in  imperfect  contacts,  it  is  very  important  to  make  all  connec- 
tions carefully  and  to  see  that  the  plugs,  particularly  of  the  resistance  box 
R2,  are  well  cleaned  with  benzine  and  carefully  inserted. 

Reverse  the  direction  of  the  current  through  the  galvanometer  and 
repeat.  Read  the  P.D.  on  the  voltmeter  at  the  time  of  taking  the  readings 
of  the  deflections.* 

Calculate  the  value  of  K  for  each  value  of  7?2,  using  in  each  case  the 
mean  deflection  for  d.  Neglect  in  the  calculation  any  figures  the  dropping 
of  which  from  the  result  will  not  introduce  into  the  value  of  K  an  error 
larger  than  that  due  to  the  observational  error  in  reading  the  voltmeter. 
Express  K  in  amperes  per  radian  and  also  in  absolute  electro-magnetic 
units  per  radian.  Record  also  the  "  figure  of  merit "  and  the  "  sensibility" 
of  the  galvanometer. 

EXAMPLE 

The  galvanometer  was  found  to  have  a  resistance  of  536.8  ohms.  The 
resistance  Rl  was  made  20,000  ohms,  and  Rs  1000  ohms.  When  R2  was 
1  ohm  the  average  deflection  for  direct  and  reversed  currents  was  3.10  cm. 
on  a  scale  distant  143.4  cm.  from  the  galvanometer  mirror.  As  the  volt- 
meter reading  across  the  terminals  of  the  Daniell  cell  used  was  1.05  volts, 
this  deflection  corresponded  to  a  current  of  .0000000342  ampere.  Hence 
K  was  .00000315  ampere.  Similar  readings  for  R2  =  2  and  R2  =  3  ohms 
gave  6.3  cm.  and  9.48  cm.  as  values  of  the  deflection,  and  .00000310  and 
.00000309  for  the  corresponding  values  of  K.  The  average  value  of  K 
was  therefore  .00000311  ampere,  or  .000000311  absolute  unit  of  cur- 
rent. The  figure  of  merit  of  the  galvanometer  was  1.55  x  10~9,  and  the 
sensibility  645  megohms. 

*  If  the  voltmeter  is  in  demand  it  may  be  freed  from  this  experiment  as 
follows.  Observe  carefully  one  deflection  when  the  voltmeter  is  connected  as 
in  Figure  62,  then  disconnect  the  voltmeter.  If  the  deflection  changes,  the  P.D. 
at  the  terminals  of  the  cell  when  the  voltmeter  is  disconnected  is  the  voltmeter 
reading  multiplied  by  the  ratio  of  the  second  deflection  to  the  first.  Thenceforth 
use  the  cell  without  the  voltmeter,  but  use  in  the  calculations  this  corrected 
value  of  the  P.D. 


CHAPTER  VIII 
THE   ABSOLUTE   MEASUREMENT   OF   CAPACITY 

58.  Definition  of  capacity.  The  difference  of  electrical  poten- 
tial between  any  conductor  and  the  earth  is  commonly  called 
simply  the  potential  of  the  conductor  and  is  designated  by  the 
letter  V.  In  other  words,  the  electrical  potential  of  the  earth  is 
arbitrarily  chosen  as  the  zero  from  which  the  potentials  of  all 
other  conductors  are  measured. 

It  follows  from  this  convention,  and  from  the  definition  of  P.I), 
given  on  page  11,  that  the  electrical  potential  in  absolute  units  of 
any  conductor  A  (Fig.  67)  on  which  there  is  a  charge  of  Q  units 
^_^  of  electricity  is  equal  to  the  number  of 

(+)  ergs  of  work  required  to  carry  unit  charge 

A  of  positive  electricity  from  the  earth  E  up 

to  A  against  the  force  which  the  charge  Q 
exerts  upon  this  unit  charge. 

67  Suppose  now  that  the  quantity  Q  were 

to  be  doubled.    The  field  strength  at  all 

points  between  the  earth  and  A  would  obviously  be  doubled  -also, 
and  hence  the  work  required  to  carry  unit  charge  up  to  A  would 
be  doubled.  In  other  words,  under  the  conditions  indicated,  the 
ratio  between  the  charge  on  A  and  the  potential  which  this  charge 
imparts  to  A  is  a  constant. 

This  constant  ratio  between  the  charge  on  a  conductor  and  its 
potential  is  called  the  electrical  capacity  of  the  conductor,  and  is 
denoted  by  the  letter  C.  Thus  for  the  defining  equation  of  capacity 

we  may  write     pharo.p  n 

Charge    =  Q  = 

Potential  V 

We  may  state  the  definition  in  words  as  follows :  The  capacity  of 
any  conductor  is  the  charge  which  must  be  placed  upon  it  in  order 
to  raise  its  potential  one  unit  above  the  potential  of  the  eartli. 

90 


ABSOLUTE  MEASUREMENT  OF  CAPACITY'          91 

59.  Factors  upon  which  capacity  depends.  A  charge  placed 
upon  an  isolated  conducting  sphere  establishes  at  any  point  out- 
side the  sphere  a  held  intensity  of  the  same  value  as  would  be 
produced  at  that  point  were  the  whole  charge  concentrated  at  the 
center  of  the  sphere.  If,  then,  the  size  of 
the  sphere  A  is  increased  while  the  charge 
upon  it  is  kept  constant,  the  unit  charge 
which  we  are  imagining  to  be  brought  up 
from  the  earth  to  A  does  not  need  to  ap- 
proach so  near  to  the  center  of  the  charge 
A  hi  order  to  be  placed  upon  the  sphere, 
and  therefore  does  not  require  as  great  an 

amount  of  work.  Indeed  it  may  be  shown,  by  a  more  extended 
analysis  than  will  be  attempted  here,  that  the  amount  of  work 
required  varies  inversely  as  the  radius  of  the  sphere.  Since 
the  amount  of  work  required  represents  the  potential  of  the 
sphere,  it  follows  that  the  potential  of  an  isolated  conducting 
sphere  carrying  a  given  charge  varies  inversely  as  its  radius,  and 
hence  that  the  capacity  of  a  sphere  varies  directly  as  its  radius. 

A  second  and  very  important  factor  upon  which  the  capacity 
of  a  conductor  may  depend  is  the  presence  of  neighboring  con- 
ductors. Thus  suppose  that  a  negatively  charged  body  B  is 
placed  near  A,  as  in  Figure  68.  It  is  clear  that  the  work 
required  to  bring  a  unit  positive  charge  from  the  earth  to  A  will 
now  be  less  than  when  A  was  isolated,  for  the  attraction  which 
the  negative  charge  on  B  exerts  upon  the 
unit  positive  charge  which  we  are  imag- 
ining to  be  brought  from  the  earth  to  A 
will  partially  neutralize  the  repulsion  due 
to  A.  Hence  the  capacity  of  A,  i.e.  the 
charge  required  to  raise  it  to  a  given 

"•*"*"  potential,  is  much  greater  when  B  is  near 

.b  IG.  69 

to  it  than  when  B  is  remote. 

If  the  body  B  is  not  initially  charged,  but  is  simply  connected 
to  earth  by  a  conductor  as  in  Figure  69  and  then  brought  near 
to  A,  it  will  acquire  a  charge  by  electrostatic  induction  (see  sect.  5, 
p.  4).  That  is,  A  will  induce  upon  B  a  charge  opposite  in  kind 


92  ELECTKICITY,  SOUND,  AND  LIGHT 

to  that  upon  itself  by  attracting  toward  itself  an  unlike  kind  of 
electricity  and  by  repelling  a  like  kind  along  the  wire  toward  the 
earth.  The  conductor  B  will  therefore  lower  the  potential  and 
thus  increase  the  capacity  of  A,  just  as  well  as  though  the  negative 
charge  had  been  imparted  to  it  before  it  was  brought  near  to  A. 
In  other  words,  the  capacity  of  a  conductor  may  be  very  greatly 
increased  ~by  simply  bringing  near  it  another  conductor  which  is 
connected  to  earth.  On  account  of  this  fact  two  conductors  which 
are  very  close  together  and  so  arranged  that  one  of  them  can  be 
connected  to  earth  are  said  to  constitute  an  electrical  condenser. 

60.  The  condenser.  It  will  be  clear  from  the  above  discussion 
that  if  we  wish  to  increase  enormously  the  capacity  of  a  conductor 
by  the  presence  of  an  adjacent  conductor,  we  have  only  to  give  to 
the  two  conductors  such  shapes  that  one  can  be  brought  almost 

into  coincidence  with  the  other.  This 
condition  is  realized  by  making  the  con- 
ductors thin  metallic  sheets,  or  plates, 
which  are  separated  from  each  other  by 
a  very  thin  layer  of  air,  or  other  insu- 
lating material.  Hence  condensers  usu- 
ally  take  the  form  of  two  sheets  of  tin 
foil  separated  by  thin  sheets  of  mica,  or 

glass,  or  paraffined  paper.  To  charge  such  a  condenser  we  might 
connect  one  plate  B  to  earth  and  join  the  other  plate  A  to  one 
terminal  of  any  electrical  generator  6,  for  example  a  static  machine 
or  a  galvanic  cell.  But  if  we  wished  to  raise  A  to  as  high  a  poten- 
tial as  possible  above  the  earth,  we  should  also  connect  the  other 
terminal  of  the  generator  to  earth  (Fig.  70),  for  then  the  full  differ- 
ence of  potential  which  the  generator  is  able  to  maintain  would 
be  built  up  between  the  earth  and  A.  Furthermore,  since  B  is 
also  connected  to  earth,  its  potential  is  necessarily  zero,  i.e.  the 
same  as  that  of  the  earth,  since  all  points  on  a  conductor  in  a  static 
condition  are  always  at  the  same  potential  (p.  12).  Hence  the 
capacity  of  A  in  this  condition  would  be  simply  the  charge  upon  it 
divided  by  the  P.D.  between  A  and  B,  i.e.  by  the  amount  of  work 
required  to  carry  unit  charge  from  B  to  A.  But  it  is  clear  that  we 
could  produce  the  same  P.D.  between  A  and  B  by  connecting  the 


ABSOLUTE   MEASUREMENT  OF  CAPACITY          93 

terminals  of  the  generator  directly  to  A  and  B  without  the  inter- 
vention of  the  earth,  for  in  either  case  the  P.D.  between  A  and  B 
would  be  the  total  number  of  volts  which  the  generator  is  able  to 
maintain.  Hence  it  is  customary  to  charge  a  condenser  by  con- 
necting it,  as  in  Figure  71,  with  a  battery  b  and  pressing  the  key  kr 
It  is  evident  that  the  two  opposite  charges  upon  A  and  B  are  neces- 
sarily of  the  same  size,  since  A  and  B  are  simply  the  terminals 
of  the  generator  &,  and  since  positive  and  negative  electricities 
always  appear  in  exactly  like  amount  whatever  be  the  means  by 
which  they  are  generated.  We  have,  then,  the  charge  Q  upon  either 
plate  A  or  B  divided  by  the  P.D.  between  A  and  B  as  the  capacity 
of  the  condenser.  We  can  obtain  the  P.D.  at  once  with  a  volt- 
meter attached  to  the  terminals  of  the  cell  b.  To  obtain  the  charge 
Q  upon  either  plate, 
we  discharge  the  | 

-L/ 

the  galvanometer  G. 
This  is  done  by  open- 


condenser  through 


ing    kl    and    closing 
ky  We  may  conceive 

of   the  discharge   as  FIG.  71 

being     accomplished 

either  by  the  passage  of  the  total  charge  on  A  around  to  B  (this 
would  accord  with  Franklin's  one-fluid  theory,  which  made  nega- 
tive the  absence  of  positive),  or  by  the  passage  of  the  total  charge 
on  B  around  to  A  (this  would  accord  with  the  electron  theory, 
which  makes  the  negative  the  mobile  kind  of  electricity),  or  by 
the  passage  of  half  of  the  charge  on  A  around  to  B  and  half  of 
that  on  B  around  to  A  (this  would  be  in  accordance  with  the  old 
two-fluid  hypothesis).  In  any  case,  however,  the  total  quantity 
which  would  pass  through  the  galvanometer  would  be  the  total 
charge  Q  on  one  of  the  plates. 

61.  To  find  Q  from  the  throw  9  of  the  galvanometer,  the  gal- 
vanometer constant  K,  and  the  period  t  of  its  half  vibration. 
While  the  quantity  Q  passes  through  the  coil  of  the  D'Arsonval 
galvanometer,  it  constitutes  an  electrical  current  which  reacts 
upon  the  magnetic  field  of  the  galvanometer  so  as  to  impart 


94  ELECTRICITY,  SOUND,  AND  LIGHT 

an  impulse  to  the  coil.  If  i  represent  the  mean  value  of  this 
current,  then  the  mean  moment  of  force  Fh  acting  on  the  coil 
while  this  current  is  flowing  is  given  by 

~Fh  =  i&CA     (eq.  (5),  p.  81).  (2) 

If  r  represents  the  time  during  which  this  discharge  takes  place, 
then  the  total  impulse  acting  on  the  coil,  i.e.  the  moment  of  force 
times  the  time,  is  given  by 


Fhr  =  i-rWA  =  QfflA.  (3) 

Now,  by  Newton's  second  law,  the  impulse,  or  product  of  the  force 
by  the  time  during  which  it  acts,  is  equal  to  the  momentum 
imparted  (ft  =  mat  =  mv).  Similarly,  in  rotation,  the  moment  of 
an  impulse,  or  the  impulse  multiplied  by  its  lever  arm,  is  equal  to 
the  moment  of  momentum,  or  angular  momentum,  imparted. 

If,  then,  /  represents  the  moment  of  inertia  of  the  coil  and  co  the 
angular  velocity  imparted  to  it  by  the  impulse,  we  have 

Jkr  =  Ito.  (4) 

Furthermore,  we  have  seen  (p.  81)  that  the  constant  of  the  gal- 
vanometer K  is  equal  to  T0/A&C.  We  get,  then,  by  substitution 
of  these  new  values  of  Fh  r  and  &CA  in  (3), 

/*>  =  £§       or       Q  =  K^~.  (5) 

A  J.Q 

Since  we  do  not  observe  directly  the  initial  angular  velocity  « 
of  the  coil,  but  rather  the  angular  distance  6  through  which  it 
moves  because  of  this  velocity,  we  must  find  a  way  of  expressing 
&)  in  terms  of  6.  This  is  most  easily  done  by  equating  the  initial 
kinetic  energy  of  the  coil,  namely  |-/<w'2,  to  the  work  done  in 
moving  this  coil  an  angular  distanced  against  the  torsional  resist- 
ance of  the  suspension.  The  value  of  this  resistance  at  the  end  of 
the  swing,  i.e.  when  the  deflection  is  6,  is  T00,  in  which  T0  is  the 
moment  of  torsion  of  the  suspension.  Since  the  resistance  of  tor- 
sion is  proportional  to  the  displacement,  the  mean  resistance  of 


ABSOLUTE  MEASUREMENT  OF  CAPACITY 


95 


torsion  while  the  coil  is  swinging  through  the  angle  6  is  \  T^O. 
The  work  done  against  torsion  is  the  resistance  times  the  distance. 
We  have,  then, 


or 


Substituting  in  (5)  we  obtain 


Now  the  half  period  t  of  any  torsioiial  system  is  given  by 

* 


(6) 


Substituting  this  value  of  I/T0  in  (5)  we  obtain 

K0t 


7T 

This  gives  us  Q  in  terms  of  the  very  easily  observed  quantities  6 
the  throw,  K  the  galvanometer  constant,  and  t  the  half  period  of 
vibration  of  the  suspended  coil. 

62.  Correction  of  the  formula  for 
damping.  In  the  above  deduction  of 
the  relation  between  o>  and  6  we 
tacitly  assumed  that  the  vibration 
of  the  galvanometer  coil  takes  place 
altogether  without  frictional  losses  of 
any  sort ;  that  is,  we  assumed  an  un- 
damped swing.  The  damping  which 
in  fact  always  exists  may  be  taken 
into  consideration  as  follows :  If  the 
line  oo'  (Fig.  72)  represents  the  posi- 
tion of 'rest  of  the  galvanometer,  and 
if  the  amplitudes  0lt  02,  08,  etc.,  of 
successive  swings  are  represented  by 

the  distances  of  the  successive  turning  points  from  the  line  oo' , 
then  it  is  found  by  experiment  that  the  law  which  always  holds, 


FIG.  72 


*  See  "  Mechanics.  Molecular  Physics,  and  Heat,"  pp.  87-91. 


96  ELECTRICITY,  SOUND,  AND  LIGHT 

approximately  at  least,  for  any  damped  vibration  is  that  the  suc- 
cessive amplitudes  bear  to  one  another  a  constant  ratio.  If  we 
call  this  ratio  p,  we  have 


That  is,  01  =  p02,         02  -  p0z,  etc. 

or,  in  general,  0m  =  pn~m0n.  (8) 

This  equation  tells  us  that  any  amplitude  0m  may  be  obtained 
from  any  later  amplitude  0n  by  multiplying  6n  by  the  damping 
factor  p  raised  to  a  power  corresponding  to  the  number  of  swings 
through  which  the  damping  acts  between  6m  and  0n. 

In  accordance  with  this  rule  the  amplitude  6,  which  would 
have  been  attained  in  the  first  swing  if  there  had  been  no  damp- 
ing, may  be  obtained  from  the  actual  amplitude  0l  of  the  first 
swing  by  multiplying  01  by  p  raised  to  the  power  corresponding 
to  the  number  of  swings  through  which  the  coil  has  been  damped 
between  the  instant  at  which  it  starts  and  the  instant  at  which  it 
reaches  0r  Since  this  is  one  half  swing,  we  have  6  =  pWr  The 
value  of  p  may  be  found  by  observing  any  two  amplitudes,  say 
the  first  and  the  twenty-fifth,  and  then  substituting  in  equation 
(8),  which,  for  this  case,  becomes 

\  a 

(9) 

We  obtain,  then,  as  the  final  form  of  the  equation  for  the  determina- 
tion of  the  quantity  Q  in  terms  of  the  throw  of  the  galvanometer, 

(10) 


7T 


If,  then,  V  denotes  the  potential  difference  between  the  plates 
and  0  the  capacity  of  the  condenser,  we  have 


ABSOLUTE  MEASUREMENT  OF  CAPACITY          97 

63.  The  ballistic  galvanometer.    It  is  evident  that  we  cannot 
determine  the  capacity  of  a  condenser  by  the  above  method  unless 
we  use  a  galvanometer  which  oscillates  for  a  long  time,  that  is, 
one  which  does  not  damp  down  rapidly ;  for  t  cannot  be  deter- 
mined accurately  unless  a  considerable  number  of  swings  can  be 
observed.    Furthermore,  the  damping  law  mentioned  above  obvi- 
ously cannot  hold  for  a  very  rapid  rate  of  damping,  for  if  the  sys- 
tem should  not  make  more  than  one  or  two  swings,  we  evidently 
could  not  say  that  the  ratio  of  successive  swings  was  constant. 

Now  a  galvanometer  which  is  designed  to  reduce  damping  to  a 
minimum  is  called  a  ballistic  galvanometer.  The  only  differences 
between  a  ballistic  and  a  nonballistic  D' Arson val  galvanometer  lie 
in  the  absence  from  the  former  of  all  mechanical  damping  devices, 
and  in  the  fact  that  the  coil  is  not  wound  on  a  conducting  frame, 
for  such  a  frame  causes  electro-magnetic  damping.* 

64.  Units   of  capacity.    If  Q  and  V  are  measured  in  absolute 
electro-magnetic  units,  then  C  will  also  be  obtained  in  absolute 
units ;  but  if  Q  is  measured  in  coulombs  and  V  in  volts,  then  C 
will  be  obtained  in  practical  units.    The  practical  unit  of  capacity 
is  named  the  farad  in  honor  of  Faraday.    It  is  the  capacity  of  a 
condenser  which  acquires  a  P.D.  of  1  volt  when  it  receives  a  charge 
of  1  coulomb.    Thus 

.      ,        coulombs      10'1 

farads  =  —  —  =  —  —  =  10~9  absolute  units.          (12) 

volts  1 08 

The    farad  is   so  large   a  unit  that  the  microfarad   (=  .000001 
farad)  is  the  unit  which  is  now  most  commonly  in  use. 

EXPERIMENT  8 

Object.  To  make  an  absolute  measurement  of  the  quantity  of  electricity 
discharged  by  a  condenser,  charged  to' a  known  difference  of  potential,  and 
hence  to  determine  the  capacity  of  the  condenser. 

Directions.  I.  Set  up  in  the  manner  indicated  in  Figure  73  a  standard 
condenser  C  (between  .1  and  1.  microfarad),  a  Daniell  cell  B,  a  voltmeter 

*  The  conducting  frame  rotating  in  the  magnetic  field  of  the  galvanometer 
would  have  induced  in  it  a  current  in  such  a  direction  as  to  oppose  its  motion. 
See  Chapter  XII. 


98 


ELECTRICITY,  SOUND,  AND  LIGHT 


F,*  a  ballistic  D'Arsonval  galvanometer  G,  a  commutator  c,  a  damping  key 
A-.2,  and  a  discharge  key  kr    This  last  instrument  is  merely  a  special  form 

of  two-point  key  (see  Fig.  74). 
It  is  best  to  connect  it  so  that 
when  the  tongue  is  pressed  down 
the  battery  will  charge  the  con- 
denser through  the  lower  con- 
tact points  and  when  the  tongue 
is  released  the  condenser  will 
discharge  through  the  galva- 
nometer by  way  of  the  upper 
contact  points.  If  a  commuta- 
tor is  not  available,  commu- 
tate  by  interchanging  the  bat- 
tery terminals  at  d  and  e  (see 
Fig.  73). 

II.  See  that  the  scale  upon 
which  you  observe  deflections 
is  set  normal  to  the  line  of 

Fi<;.   73  ^_^y       sight,  then  bring  the  galvanom- 

eter quite  to  rest  by  closing  £2. 

If  a  swing  of  a  few  millimeters  occurs,  because  of  thermal  effects,  when 
A*2  is  opened,  it  will  be  easy  with  a  little  practice  to  press  k2  tempo- 
rarily at  such  instants  as 
to  check  the  slight  remain- 
ing swing  and  bring  the 
coil  to  rest  even  when  kz 
remains  open.  Record  this 
point  as  the  true  zero,  then 
charge  the  condenser  by 
depressing  the  tongue. 
Discharge  by  releasing  it 
and  observe  the  throw  in 
millimeters.!  Reverse  the 
commutator  and  repeat. 
The  mean  of  six  throws, 
three  to  right  and  three  to 


FIG.  74 


*  As  in  Experiment  7,  the  voltmeter  need  not  be  kept  attached  to  the  battery 
throughout  the  experiment,  provided,  in  calculating  capacity  (equation  11),  the 
voltmeter  reading  is  multiplied  by  the  ratio  between  two  throws  taken,  one 
with  the  voltmeter  disconnected,  the  other  with  it  connected. 

t  If  it  is  found  too  tedious  to  bring  the  coil  exactly  to  rest  when  &2  is  open, 
the  throw  can  be  taken  quite  accurately,  even  when  there  are  one  or  two  milli- 
meters of  swing,  in  the  following  way.  Observe  the  middle  point  about  which 


ABSOLUTE  MEASUREMENT  OF  CAPACITY          99 

left,  divided  by  twice  the  distance  from  the  scale  to  the  mirror,  should  be 
taken  as  the  correct  value  of  0  in  radians.  Since  the  zero  may  be  slightly 
variable,  a  new  zero  reading  should  be  taken  for  every  throw.  Further,  if 
the  first  observations  of  the  deflections  do  not  agree  closely,  record  no 
readings  until  successive  readings  in  a  given  direction  do  agree.* 

III.  To  obtain  ?,  discharge  the  condenser  again,  and  with  a  stop  watch 
take  the  time  of  at  least  20  half  swings,  timing  your  counts  at  the  instants  of 
passage  of  the  coil  through  its  zero  position,  not  through  its  end  positions. 

IV.  To  obtain  p  bring  the   galvanometer   again  to  rest,  charge  and 
discharge  the  condenser,  note  carefully  the  first  throw,  and,  calling  this 
first  turning  point  1,  the  next  2,  etc.,  count  the  turning  points  (the  readings 
corresponding  to  them  need  not  be  taken)  until  the  amplitude  has  been 
reduced  to  about  one  third  its  first  value,  then  note  carefully  that  ampli- 
tude.   Calculate  p  from  equation  (9). 

V.  Take    K  from   Experiment  7,  or  from  a  value    furnished  by   the 
instructor.    Calculate  in  microfarads  the  value  of  the  capacity  of  the  con- 
denser used,  and  compare  with  the  value  marked  upon  the  condenser. 

EXAMPLE 

The  condenser  used  was  charged  to  a  potential  of  V—  1.06  volts  as 
given  by  the  voltmeter  reading.  When  discharged  through  the  galvanom- 
eter it  caused  a  mean  deflection  of  15.58  cm.  on  a  scale  at  a  distance  of 
143.4  cm.  Reversing  the  direction  of  charging,  an  average  throw  of  15.56 
cm.  was  observed.  Hence  0  =  .05430  radians.  The  time  of  20  half  vibra- 
tions was  found  to  average  125.8  seconds,  hence  t=  6.29  seconds.  The 
damping  factor  p  was  found  by  observing  02  —  15.7  and  022  =  4.5.  Hence 

p  —  \\-~~  =  1.064.      .-.  pi  =1.031.      The    galvanometer    constant    K   as 
\  4.o 

found  in  Experiment  7  was  .00000311  ampere.    Hence 
c_  .00000311  x6.29  x  .05430  x  1.031 

Ti-1.06 
=  .000000329  farad  =  .3293  microfarad. 

The  condenser  used  was  marked  1  microfarad  by  the  manufacturer, 
hence  the  per  cent  of  difference  was  1.2.  (This  error  is  not  larger  than  the 
uncertainty  in  the  calibration  of  most  voltmeters.) 

the  oscillation  takes  place  and  call  this  the  zero  reading.  Then  discharge  the 
condenser  at  the  instant  at  which  the  coil  is  passing  through  one  of  its  posi- 
tions of  rest  at  the  ends  of  its  swing.  The  deflection  will  then  be  the  difference 
between  the  zero  and  the  extreme  reading  produced  by  the  discharge. 

*  The  reason  for  this  disagreement  is  that  frequently  the  suspending  wire 
acquires  a  "  set"  or  tendency  to  a  greater  deflection  in  one  direction  than  in 
the  other.  Taking  several  throws  in  the  same  direction  will  in  general,  however, 
result  in  constancy  for  that  direction  of  throw. 


CHAPTEK   IX 

COMPARISON  OF   CAPACITIES,    THE   DETERMINATION   OF 

DIELECTRIC   CONSTANTS,   AND   THE   RATIO  OF  THE 

ELECTROSTATIC  AND  ELECTRO-MAGNETIC  UNITS 

65.  Comparison  of  capacities  by  means  of  a  ballistic  galva- 
nometer.   From  the  definition  of  capacity  (in  symbols,  C  =  Q/PD) 
it    follows    that    the    quantities    of    electricity   acquired   by   two 
separate  condensers  when  charged  to  the  same  P.D.  vary  directly 
as  their  capacities.    Since  the  throws  of  a  ballistic  galvanometer 
are  proportional  to  the  quantities  of  electricity  passing  through  it, 
the  ratio  between  the  capacities  of  two  condensers  may  be  found 
by  charging  them   to  the  same  P.D.  and  noting  the  ratio  of  the 
throws  they  cause  whqn  discharged  through  the   same  ballistic 
galvanometer.    If  one  of  the  condensers  is  a  standard  of  known 
capacity,  the  value  of  the  capacity  of  the  second  condenser  may 
be  found  from  this  ratio. 

66.  The  bridge  method  of  comparing  capacities.    A  comparison 
of  capacities  may  also  be  made  by  a  method  which  is  analogous 
to  the  Wheatstone-bridge  method  of  comparing  resistances,  i.e.  a 
method  in  which  a  condition  of  balance  is  indicated  by  no  deflec- 
tion of  the  galvanometer  (a  so-called  "  zero  method ").    The  two 
condensers  to  be  compared  are  connected  as  in  Figure  75.    Cl  and 
Cz  represent  their  capacities.    R  and  S  are  two  resistance  boxes. 
The  key  K  permits  of  the  charging  or  discharging  of  the  condensers. 
A  galvanometer  is  connected  between  a  and  c.    When  the  galva- 
nometer shows  no  deflection   on   charging  or  discharging,  there 
exists  a  condition  of  balance  for  which,  as  is  shown  in  the  next 
paragraph,  the  following  relation  holds : 

/*T  Of 

£1  =  .^. 

Cz      R 

100 


COMPARISON  OF  CAPACITIES; 

Since  by  the  assumption  of  no  current  through  the  galvanom- 
eter the  points  a  and  c  are  always  at  the  same  potential,  it  follows 
that  during  the  whole  time  of  discharge,  for  example,  the  P.D. 
between  b  and  a  is  always  equal  to  that  between  b  and  c.  Now 
it  follows  from  Ohm's  law  that  if  the  P.D.  between  b  and  a  is  the 
same  as  that  between  b  and  c,  the  currents  flowing  hi  the  branches 
R  and  S  are  inversely  proportional  to  the  resistances  of  R  and  S. 
Since,  however,  the  time  during  which  the  condensers  are  discharg- 
ing is  the  same  for  both,  —  for  otherwise  a  current  would  have  to 


FIG.  75 


flow  through  the  galvanometer,  —  it  follows  that  the  quantities  dis- 
charged through  R  and  S  by  the  condensers  1  and  2  are  propor- 
tional to  these  currents,  and  hence  inversely  proportional  to  the 
resistances  R  and  S.  That  is, 

§=!•  (1) 

Q2     R 

But  since  a  and  c  are  always  at  the  same  potential,  the  condensers 
must  always  be  charged  to  the  same  P.D.,  and  hence  the  quantities 
which  they  hold  must  be  proportional  to  their  respective  capacities. 
Substituting  for  QjQ^  the  equal  ratio  CJCZ  we  obtain,  as  the  con- 
dition which  must  exist  for  no  galvanometer  deflection, 

£  =  4-  (2) 


102 


,  SOUND,  AND  LIGHT 


67.  The  bridge  method,  using  an  induction  coil.  No  essential 
difference  is  introduced  if  the  charging  and  discharging  of  the 
condensers  in  the  method  just  described  is  accomplished,  not  by 
the  key,  but  by  the  use  of  an  alternating  current.  To  this  end 
the  points  b  and  d  in  Figure  75  are  connected  to  the  secondary  of 
an  induction  coil,  to  the  primary  of  which  is  connected  the  battery. 
The  effect  of  the  automatic  make  and  break  of  the  current  in  the 
primary  circuit  is  to  set  up  in  the  secondary  a  current  increasing 
and  decreasing  in  alternate  directions.  This  charges  the  condensers, 
permits  their  discharge  as  it  decreases,  and,  recharging  in  an  oppo- 

site direction,  again  permits  their 
discharge.  If  the  balance  is  not 
perfect,  there  will  be  a  rapidly 
alternating  P.D.  between  a  and  c. 
In  the  place  of  the  galvanometer 
a  telephone  receiver  is  therefore 
used,  and  the  test  for  a  balance 
becomes  the  absence  of  a  buzz- 
ing sound  in  the  telephone.* 

68.  The  dielectric  constant. 
When  the  capacities  of  con- 
densers which  differ  from  one 
another  only  in  the  nature  of 
the  medium  between  the  plates  are  compared  by  either  of  the 
methods  given  above,  it  is  found  that  they  differ  widely.  Hence 
Faraday  introduced  the  term  "  specific  inductive  capacity  "  to  denote 
the  constant  of  the  medium  which  is  measured  by  the  ratio  between 
the  capacity  of  a  condenser  which  has  the  given  medium  between  its 
plates  and  the  capacity  of  the  same  condenser  when  air  (or  more 
strictly  a  vacuum}  is  betiveen  the  plates.  Thus  the  specific  in- 
ductive capacity  of  ordinary  glass  is  from  2.3  to  2.8,  that  of  hard 
rubber  is  from  2  to  3,  that  of  paraffin  1.8  to  2.3,  that  of  mica  2  to  4. 
This  constant  relates,  of  course,  only  to  insulators  or  dielectrics. 
Hence  it  is  now  commonly  called  the  dielectric  constant. 

*  Precisely  the  same  equations  as  were  developed  in  section  66  hold,  not  only 
for  the  scheme  of  connections  mentioned  in  this  section,  but  also  for  that  shown 
in  Figure  76,  where  the  telephone  Tand  the  induction  coil  have  been  iiitercharged. 


FIG.  76 


DIELECTRIC  CONSTANT  103 

69.  Dielectric  absorption.    The  results  of  a  comparison  of  two 
condensers  by  the  bridge  method,  described  in  section  67,  will  often 
not  agree  closely  with  the  results  of  a  comparison  by  the  method 
of  ballistic  throws.    Particularly  is  this  true  if  the  condensers  com- 
pared are  made  of  quite  different  insulating  materials,  e.g.  mica 
and  paraffin.    The  reason  for  such  a  disagreement  is  as  follows. 

The  capacity  of  a  condenser  has  been  denned  as  the  constant 
ratio  between  the  charge  given  it  and  the  P.D.  to  which  it  is 
charged.  But  when  we  measure  a  capacity  by  finding  the  ratio  of 
the  quantity  discharged  to  the  charging  P.D.,  we  tacitly  assume  the 
equality  of  the  quantity  of  charge  and  that  of  discharge.  As  the 
result  of  a  phenomenon  known  as  dielectric  absorption,  such  an 
equality  does  not  always  exist.  The  insulating  medium  between 
the  condenser  plates,  called  the  dielectric,  appears  to  absorb  a 
portion  of  the  charge.  Thus  if  a  discharged  condenser  be  allowed 
to  stand  for  a  short  time  on  open  circuit,  a  second  or  even  a  third 
charge  may  be  taken  from  it.  This  residual  charge  is  practically 
negligible,  however,  in  the  case  of  the  ordinary  mica  condenser. 
The  amount  of  the  absorption  for  any  given  condenser  is  some  com- 
plicated function  of  the  time  consumed  in  charging  and  discharging. 
Methods  of  comparison,  therefore,  which  employ  markedly  different 
times  will  not  in  general  agree  closely. 

70.  Laws  for  the  combination  of  condensers.    Using  either  of 
the  methods  of  comparison  described  above,  it  is  possible  to  test 
the  laws  for  the  combination 


of  condensers.    These  laws         _j_Cy       Q — -^2.     ^  ^[_3 
follow  at  once  from  the  de- 


nning equation  of  capacity. 


Thus    in   Figure    77   are 

shown  three  condensers  of  capacities  Clt  C2,  and  C3  respectively, 
connected  in  parallel.  Obviously  they  are  all  charged  to  the  same 
potential.  By  definition,  the  quantities  they  receive,  Qv  Q2,  and  Qs, 
are  then  expressed  in  terms  of  their  common  P.D.  as  follows : 

Q,  =  C,(PD),   Q.2  =  C2(PD),   QS  =  C,(PD).  (3) 

The  total  charge  Q  which  they  receive  is  the  sum  of  these  separate 
charges.  That  is,  Q  =  Q,+ Qz+ Qy  (4) 


104  ELECTRICITY,  SOUND,  AND  LIGHT 

Now  if  C  represents  their  joint  capacity,  Q=  C(PD),  and  there- 
fore, by  substitution  in  (4)  of  the  values  of  Q,  Qv  Q2,  and  Qs,  we 

obtain 


That  is,  for  condensers  in  parallel  the  joint  capacity  is  the  sum  of 
the  several  capacities. 

In  Figure  78  are  shown  these  same  condensers  connected  in 
series.  PDV  PD2,  and  PD3  represent  their  respective  differences 
of  potential.  In  this  case  the  total  potential  difference  PD 
between  the  ends  of  the  series  is  necessarily  equal  to  the  sum 
of  the  several  P.D.'s.  .  That  is, 


PD  =  PD^  +  PD2  +  PD3.  (6) 

The  quantities  of  electricity  held  by  all  these  condensers  are  the 
same.  For  obviously  the  +  charge  011  condenser  3  equals  the 
—  charge  on  condenser  1,  since  the  generator  B  must  develop 

equal  amounts  of  positive  and 
negative  electricity.  But  since 
the  +  charge  and  the  —  charge 
on  the  plates  of  a  condenser 


PDo  PO,. 

Ill 


7  V  °3 


TB  r 

are  equal,  the  — charge  on  con- 
denser 1  equals  the  +  charge 
on  the  opposite  plate.  And  this 

+  charge  is  the  same  as  the  —  charge  on  the  plate  of  condenser  j£, 
for  the  4-  charge  011  condenser  1  and  the  —  charge  on  condenser  2 
are  simultaneously  produced  by  electrostatic  induction,  and,  accord- 
ing to  the  laws  of  induction  as  discovered  by  Faraday,  +  and 
—  charges  always  appear  in  equal  amount.  From  these  considera- 
tions it  is  clear  that 

But,  by  definition, 

Q  =  C(PD),  Q^C^PDJ,  QZ  =  CZ(PD2),  and  QS  =  CS(PD)S.  (8) 


By  substitution  in  (6)  of  the  values  of  PD,  PDV  P£>2,  and  PDS, 
given  in  (8),  it  follows,  in  consideration  of  (7),  that 

+- 


RATIO  OF  THE  UNITS 


105 


That  is,  for  condensers  in  series  the  reciprocal  of  the  joint  capacity 
is  the  sum  of  the  reciprocals  of  the  several  capacities.  Hence  the 
law  for  capacities  in  series  is  similar  to  the  law  for  resistances  in 
parallel,  and  vice  versa. 

71.  Standard  condensers.  Condensers  carefully  constructed  of 
tin-foil  plates  and  sheets  of  mica  as  the  dielectric  are  arranged  in 
boxes  similar  in  form  to  resistance  boxes.  In  Figure  79  is  shown 
a  standard  form  of  subdivided  condenser.  The  binding  posts  b  and 
b'  are  connected  to  the  brass  strips  M  and  N  respectively.  Between 
M  and  N,  and  capable  of  connection  by  plugs  to  either  M  or  N, 
are  other  strips  A,  B,  C,  D,  E,  and  F.  The  condensers  are  con- 
nected between  these  as  indicated  by  the  dotted  connections.  To 
connect  between  b  and  V  any  single  capacity,  for  example  that  of 
the  condenser  between  E  and  F,  we  connect  M  to  E  by  a  plug 


M 

„ 

-~  ! 

b 

9 

c 

A 

i    i 
H   }••• 

i  : 

v^- 

B 

—  *       f  

C 

-    - 

r^^- 

D 
.^-^ 

L 

^r 

E 
j~^ 

i 

-v_A 

F 

./-^ 

i 

6' 

i  

N 

' 

FIG.  79 

and  also  N  to  F.  If  it  is  desired  to  have  the  sum  of  the  capacities 
of  the  three  condensers  CD,  DE,  and  EF  between  the  binding 
posts,  since  capacities  in  parallel  are  added,  we  connect  the  con- 
densers in  parallel  by  plugging  D  and  F  to  M,  and  also  C  and  E  to  N. 

Since  the  sum  of  the  reciprocals  of  several  capacities  in  series 
is  the  reciprocal  of  their  joint  capacity,  to  obtain  a  small  fraction, 
we  connect,  for  example,  A  to  M  and  C  to  N,  thus  putting  con- 
densers AB  and  BC  in  series  between  b  and  b'. 

72.  The  ratio  of  the  electro-magnetic  and  the  electrostatic 
units.  As  stated  in  section  28  (p.  35),  the  electro-magnetic  unit  of 
quantity  is  found  to  be  equal  to  3  X  1010  electrostatic  units,  and 
the  fact  that  this  number  is  precisely  the  same  as  the  velocity  of 
light  in  centimeters  has  been  one  of  the  chief  factors  in  the  estab- 
lishment of  the  electro-magnetic  theory  of  light. 


106  ELECTRICITY,  SOUND,  AND  LIGHT 

This  relation  between  the  units  is  made  the  object  of  an  experi- 
mental verification  in  the  experiment  following  this  chapter.  It 
is  not,  however,  a  quantity  of  electricity  which  will  be  directly 
measured  in  the  two  systems,  but  rather  the  capacity  of  a  con- 
denser. From  this  latter  measurement  the  former  can  be  easily 
calculated,  as  will  appear  from  the  following  considerations.  If 
the  electro-magnetic  unit  is  in  fact  3  x  1010  electrostatic  units,  it 
follows  that  when  a  given  quantity  of  electricity  is  measured  in 
electrostatic  units  we  should  get  a  number  3  x  1010  larger  than 
when  we  measure  the  same  quantity  in  electro-magnetic  units. 
Furthermore,  since  the  unit  of  P.D.  in  either  system  is  the  P.D. 
between  two  points  when  it  requires  1  erg  of  work  to  carry  unit 
quantity,  measured  in  that  system,  between  the  points,  it  will 
be  seen  that  a  P.D.  measured  in  electro-magnetic  units  should  be 
3  X  1010  times  as  large  a  quantity  as  the  same  P.D.  measured  in 
electrostatic  units. 

Now  since  C  =  Q/PD,  it  follows  that 

C  electrostatic  Q  electrostatic          PD  electro-magnetic 

C  electro-magnetic      Q  electro-magnetic         PD  electrostatic 
=  9  x  1020. 

That  is,  if  the  ratio  of  the  electro-magnetic  unit  of  quantity  to 
the  electrostatic  unit  of  quantity  is  3  x  1010,  the  capacity  of  a 
condenser  measured  in  electrostatic  units  should  be  9  X 1020 
greater  than  its  capacity  when  measured  in  electro-magnetic 
units.  In  order,  then,  to  determine  the  ratio  of  the  units  of  quan- 
tity, we  have  only  to  measure  in  electro-magnetic  units,  by  the 
method  of  Chapter  VIII,  the  capacity  of  a  condenser  of  simple 
geometric  form  for  which  the  capacity  in  electrostatic  units  may 
be  calculated  according  to  relations  which  will  be  established  in 
the  next  section.  Since,  however,  the  capacity  of  a  condenser  of 
form  suitable  to  this  calculation  is  very  small,  we  shall  deter- 
mine its  capacity  in  electro-magnetic  units  by  comparison  with 
a  larger  condenser,  the  capacity  of  which  has  already  been 
measured  in  absolute  electro-magnetic  units  by  the  method 
discussed  in  the  preceding  chapter. 


RATIO   OF  THE   UNITS 


107 


73.  The  calculation  of  the  capacity  of  a  plate  condenser  in  elec- 
trostatic units.  We  can  compute  C  in  electrostatic  units  at  once,  if 
we  know  the  area  of  the  condenser  plates,  the  distance  between 
them,  and  the  dielectric  constant  of  the  medium.  To  see  how  this 
is  done,  let  us  first  find  the  field  strength  in  dynes  which  exists 
between  two  condenser  plates  in  air  when  each  of  these  plates  is 
charged  with  Q  electrostatic  units.  Let  a  represent  the  amount  of 
electricity  upon  each  square 
centimeter  of  each  plate.  c 
Th^s  quantity  cr  is  known 
as  the  density  of  charge. 
Consider  the  force  /  which 
the  charge  upon  any  little 
element  of  surface  ds  — 
(Fig.  80)  exerts  upon  a  unit 
charge  at  any  point  a  be- 
tween the  plates.  The  number  of  units  of  charge  upon  ds  is  ads. 
If  we  represent  the  distance  of  this  charge  from  a  by  r,  then  by 
the  law  of  electrostatic  force  given  in  section  2,  page  2,  we  have 


ds 


FIG.  80 


ads 


(10) 


Now  the  component  of  this  force  which  is  normal  to  AB  is  the 
only  one  with  which  we  are  concerned,  since  it  is  evident  that, 
except  at  the  very  edges  of  the  plates,*  the  resultant  force  which 
the  whole  charge  on  AB  exerts  upon  a  unit  charge  at  a  is  directed 
along  the  normal  from  AB  toward  CD.  If  f  represents  the  value 
of  the  normal  component  of  /,  we  have 


,,      o-ds 

f  =  — y-  cos  6. 


(11) 


*  The  error  introduced  by  assuming  that  the  lines  of  electrostatic  force 
between  the  plates  are  everywhere  normal  to  the  plates,  even  at  the  very  edges, 
is  of  the  same  order  of  magnitude  as  the  ratio  of  the  distance  between  the 
plates  to  their  diameter.  In  the  experiment  which  follows  this  chapter  the 
first  distance  will  be,  perhaps,  .  1  mm.  and  the  last  20  cm.  or  30  cm.  Hence  no 
appreciable  error  will  be  here  introduced  by  assuming  the  lines  to  be  everywhere 
normal  to  the  plates. 


108  ELECTRICITY,  SOUND,  AND  LIGHT 

But  if  dsr  represents  the  projection  of  ds  upon  a  plane  normal  to  r 
and  passing  through  ds  (which  it  will  be  remembered  is  an  element 
of  infinitesimal  area),  then  we  have 


(12) 
Hence  f  =     T'  (13) 

Now  the  solid  angle  subtended  by  ds  at  the  point  a  is  by  defini- 
tion ds'/r2.  If  then  we  call  this  solid  angle  u,  we  have 

f'=(TU.  (14) 

That  is,  the  force,  normal  to  the  plate,  which  each  element  of  charge 
upon  the  plate  exerts  upon  a  unit  charge  at  any  point  between  the 
plates,  is  equal  to  the  density  of  charge  upon  the  element,  multiplied 
by  the  solid  angle  subtended  by  the  element  at  the  point  consid- 
ered. Now  if  the  density  of  charge  is  the  same  at  all  points  upon 
the  plate,  then  the  total  force  F'  which  all  the  little  elements  of 
charge  on  AB  exert  on  the  unit  charge  at  a  is  a  times  the  sum 
of  all  the  solid  angles  subtended  by  all  the  elements  of  surface 
on  AB]  i.e.  it  is  a  times  the  total  solid  angle  subtended  at  a  by 
the  whole  plate  AB.  But  since  the  distance  between  AB  and  CD 
is  very  small  as  compared  with  the  dimensions  of  these  plates, 
the  total  solid  angle  subtended  at  a  by  AB  is  practically  the 
solid  angle  subtended  by  a  hemisphere  at  its  center.  The  solid 
angle  about  a  point  is  by  definition  the  surface  of  a  sphere  of 
radius  r  about  that  point  as  a  center  divided  by  r2  ;  that  is,  it  is 

4  ?rr2 

—  —  =  4  TT.     The  solid  angle  subtended  by  a  hemisphere  at  the 

center  is  therefore  2  IT.    Hence  we  have 

f"=2iro:  (15) 

Since  the  charge  upon  the  plate  CD  exerts  a  like  force  upon  the 
unit  charge  at  a,  the  total  field  strength  F  between  the  condenser 
plates  is  given  by 


Hence,  if  d  is  the  distance  from  CD  to  AB,  the  P.D.  between  the 
condenser  plates,  i.e.  the   work   required   to   carry  unit  positive 


DIELECTRIC  CONSTANT  109 

charge  from  one  to  the  other,  is  4  iro-cL  But  if  A  is  the  total  area 
of  one  plate,  the  charge  Q  upon  it  is  ACT.  Hence  the  capacity  of 
the  condenser  (7,  which  is  by  definition  Q/PD,  is  given  hy 

A(T  A 


If  we  replace  the  air  by  a  medium  of  dielectric  constant  K,  we 
have,  by  definition  of  K, 


EXPERIMENT   9 

(A)  Object.    To  test  the  laws  for  the  combination  of  condensers  in  series 
and  in  parallel  by  the  comparison  of    several  such  combinations  with  a 
standard  condenser. 

Directions.  Using  for  C\  a  subdivided  condenser  and  for  C2  a  single 
standard  condenser  of  say  ^-microfarad  capacity,  connect  as  indicated  in 
Figure  75,  except  that  a  telephone  receiver  replaces  the  galvanometer  G 
and  a  small  induction  coil  replaces  the  discharge  key  K.  R  and  S  should 
be  resistance  boxes  capable  of  a  range  of  from  1  to  10,000  ohms.  Give  to  R 
a  fixed  value  and  vary  S  until  a  balance  is  obtained.  Approach  the  value 
of  S  which  corresponds  to  a  balance  from  the  side  of  too  large  a  resistance 
and  also  from  the  side  of  too  small  a  resistance.  Record  both  observations 
and  use  the  average  in  the  calculation.  Arrange  the  plugs  in  the  sub- 
divided condenser  box  so  as  to  make  the  following  combinations  of 
condensers  :  (1)  series  connections,  .2  and  .05  microfarad  ;  (2)  parallel 
connections  .05,  .05  and  .2  microfarad. 

Compare  these  combinations  with  the  J-microfarad  condenser.  Calcu- 
late the  joint  capacity  from  the  laws  of  combination  of  condensers  and 
compare  with  the  observed  value. 

(B)  Object.    To  determine  the  dielectric 
constant  of  mica. 

Directions.  Place  a  thin  smooth  sheet 
of  mica  between  the  metal  plates  of  the 
parallel-plate  condenser  shown  in  Figure  81. 
Determine  the  capacity  of  the  condenser  so 
formed  by  the  method  used  in  (A),  being 
careful  to  select  a  value  for  C2  which  is  as  -pIG 

near  as  possible  to  that  of  the  unknown 

(say,  .05  or  .1  microfarad).    Cut  off  three  small  pieces  (3  to  5  mm.  square) 
from  the  corners  of  the  sheet  of  mica  and,  separating  the  plates  by  these, 


110  ELECTRICITY,  SOUND,  AND  LIGHT 

form  an  air  condenser  of  the  same  thickness  as  that  of  the  mica.*  Measure 
its  capacity  in  the  same  manner  and  determine  the  dielectric  constant  from 
the  ratio  of  the  capacity  with  mica  as  the  dielectric  to  the  capacity  with 
air  as  the  dielectric. 

(C)  Object.  To  determine  the  ratio  of  the  units  of  capacity,  and  hence 
of  the  units  of  quantity,  in  the  electrostatic  and  the  electro-magnetic 
systems. 

Directions.  The  capacity  in  electro-magnetic  unit?-  of  a  parallel-plate  air 
condenser  has  already  been  determined  (Exp.  9,  (B)).  It  only  remains  to 
calculate  its  capacity  in  electrostatic  units  from  the  relation  of  equation  (17). 
Measure  carefully  the  thickness  of  the  small  mica  strips  used  in  Experi- 
ment 9,  (B)  to  keep  the  plates  separated.  Take  the  average  of  at  least  ten 
observations  for  the  value  of  d.  Measure  the  radius  of  the  condenser 
plates  and  from  this  data  calculate  the  capacity.  Determine  the  ratio  of 
the  E.S.  to  the  E.M.  value  of  this  capacity,  remembering  that  one  micro- 
farad is  10~ 15  absolute  electro-magnetic  units  (see  p.  97).  Extract  the 
square  root  of  this  ratio  to  find  the  ratio  of  the  electro-magnetic  unit  of 
quantity  to  the  electrostatic  unit  of  quantity.  Compare  this  value  with  the 
value  of  v  (the  velocity  of  light),  namely,  3  x  1010. 


EXAMPLE 

(A)  The  resistance  S  was  kept  constant  at  1000  ohms.    The  value  of 
the  standard  condenser  was  £  microfarad.    The  .05  and  .05  microfarad  in 
the  multiple  condenser  box  were  connected  in  series.    For  a  balance  R 

was  74  ohms.    Hence  C  = •  —  =  .0247.    The  calculated  value  found  by 

the  relation  —  =  — -  -\ -,  namely  .0250,  differed  from  the  observed  value 

by  1.2  per  cent.  Similar  observations  with  .05  and  .05  in  parallel  gave  for 
a  balance  R  =  1000,  ,9=301;  that  is,  C=  .1003  observed  value  and  .1000 
calculated  value,  a  difference  of  .3  per  cent. 

(B)  The  average  of  thirty  readings  of  the  thickness  of  the  mica  was 
.0112  cm.    The  average  of  ten  readings  on  the  small  strips  used  in  sepa- 
rating the  plates  for  the  air  condenser  was  .0119  cm.    Using  a  .1  micro- 
farad and  making  It  =  2000  ohms,  S  was  found  to  be  130  ohms  when  the 
condenser  contained  mica.    Therefore  Cmica  =  .00650  microfarad.    Keeping 
72  =  2000  ohms  but  using  .05  microfarad,  S  was  found  to  be  113  ohms 
when  the  condenser  contained  air.     Therefore  Cair  =  .002825  microfarad. 
The  balance  in  both  cases  was  so  exact  that  a  change  of  1  ohm  in   S 
disturbed  it.    Since  the  mica  and  air  condensers  differed  in  thickness,  it 

*  If  the  mica  plate  is  not  of  uniform  thickness,  a  correction  must  be  made 
as  is  shown  in  the  example  following  this  experiment. 


RATIO  OF  THE   UNITS  111 

was  necessary  to  increase  the  capacity  of  the  air  condenser  to  the  value 
which  it  would  have  had  if  it  had  been  of  the  same  thickness  as  the  mica. 

This  value  is  evidently  ^i  |  x  .002825,  or  .00300.    Hence  the  dielectric 

.00650 

constant  of  mica  was  found  to  be  ; =  2.17. 

.00300 

(C)  In  Experiment  9,  (B),  the  capacity  of  the  air  condenser  had  already 
been  found  to  be  .002825  microfarad  for  a  thickness  of  .0119  cm. 
The  radius  of  the  plates  was  10.9  cm.  Hence  the  area  yl=7r!0.92,  and 

S          Ti-10.92  .,  the  capacity  E.S. 

C- = -—  =  2496  electrostatic  units.     Hence  -  — r^— : 

4  ird      4  TT  .01 1 9  the  capacity  E.M. 

-=8.84xl020,    and  therefore    the    ratio    of   the    electro- 


.002825  x  10- 15 

magnetic    unit    of   quantity  to    the    electrostatic    unit  was    found   to   be 
2.973  x  1010.    This  determination  differs  from  3  x  1010  by  .9  per  cent. 


CHAPTER  X 
ELECTROMOTIVE  FORCE  AND  INTERNAL  RESISTANCE 

74.  Definition  of  electromotive  force.  Whenever  two  dissimilar 
substances  are  brought  into  contact  one  of  them  is  found  to  take 
up  a  potential  higher  than  the  other.  Thus,  if  copper  and  zinc 
are  joined  as  in  Figure  82,  the  zinc  is  found  to  acquire  a  slight 
positive  charge,  and  the  copper  a  corresponding  negative  charge. 
In  order  to  assign  a  cause  to  this  phenomenon,  it  is  customary  to 
assume  that  there  exists  at  the  surface  of  contact  of  copper  and 
zinc  an  agent  which  we  call  an  electromotive  force  (E.M.F.),  which 
lias  the  ability  to  drive  positive  electricity  from  the  copper  to  the 
zinc ;  that  is,  in  terms  of  the  electron  theory,  the  ability  to  drive 
negative  electrons  from  zinc  to  copper.  In  obedience  to  this  agent 
a  current  flows,  according  to  our  conventions  (see  p.  31),  from 
Cu  Zn  the  copper  to  the  zinc  until  the  action 

^  ,.[4.       ~      ^?    of  the  agent  is  brought  to  rest  by  the 

F      82  back  pressure,  that  is,    the  P.D.   cre- 

ated between  the  zinc  and  the  copper 

because  of  the  charges  which  they  acquire.  We  then  define  an 
electromotive  force  as  any  agent  which  is  able  to  cause  the  ap- 
pearance of  positive  and  negative  electrical  charges ;  that  is, 
any  agent  which"  is  able  to  set  up  a  P.D.  And  we  measure  the 
strength  of  an  E.M.F.  by  the  P.D.  which  it  is  able  to  maintain. 
The  relations  of  E.M.F.  and  P.I),  are  then  simply  the  relations  of 
cause  and  effect.* 

*  It  is  to  be  noted  that  in  the  above  the  phenomenon  of  contact  E.M.F. 
has  merely  been  described,  and  the  term  E.M.F.  defined.  None  of  the  present 
theories  as  to  the  nature  and  cause  of  this  phenomenon  have  as  yet  been  gen- 
erally accepted.  Helmholtz  imagined  the  phenomenon  to  be  due  to  differences 
in  the  specific  attraction  of  different  kinds  of  matter  for  the  two  kinds  of  elec- 
tricity. In  terms  of  the  electron  theory  this  view  reduces  to  a  difference  in  the 
specific  attraction  of  different  substances  for  the  free  electrons.  If,  however, 
there  is  a  larger  number  of  free  electrons  in  one  substance  than  in  another, 

112 


ELECTEOMOTIYE  FOKCE  113 

75.  Ther mo-electromotive  forces.    Perhaps  the  easiest  way  of 
showing  the  existence  of  electromotive  forces  of  contact  between 
metals  is  to  place  a  sensitive  galvanometer  in  a  circuit  consisting 
of  two  different  metals,  for  example  copper  and  zinc  (Fig.  83),  and 
then  to  heat  one  of  the  junctions.    So  long  as  the  temperature  of 
the  two  junctions  a  and  b  is  the  same,  the  E.M.F.  from  copper  to 
zinc  at  a  is  necessarily  exactly  equal  and  opposite  to  the  E.M.F. 
from  copper  to  zinc  at  b,  and  hence  no  current  can  possibly  flow 
through  the  circuit.    But  as  soon  as  one  junction  is  heated  even 
a  fraction  of  a  degree  above  the  other,  a  current  is  at  once  found 
to  appear  in  the  galvanometer. 

This  is  because  the  value  of  the  Zn 
E.M.F.  of  contact  between  any 
two  substances  changes  with  the 
temperature.  Hence  if  one  junc- 
tion undergoes  a  change  in  tem- 
perature, while  the  other  does  not, 
the  balance  between  the  E.M.F.'s 

at  the  two  junctions  is  destroyed  and  a  current  flows  through  the 
circuit  which  is  proportional  to  the  resultant  E.M.F.  of  the  circuit. 
If  this  current  flows  from  a  over  to  b  through  the  zinc,  and  from 
b  back  to  a  through  the  copper,  then  there  will  of  course  be  a 
continuous  fall  of  potential  from  a  to  b  in  the  zinc  and  from  b  to 
a  in  the  copper.  Furthermore,  the  sum  of  these  falls  in  potential 
through  the  zinc  and  copper  must  of  course  be  exactly  equal, 
numerically,  to  the  resultant  E.M.F.  in  the  circuit,  for  otherwise 
there  would  be  either  a  cause  (an  E.M.F.)  without.an  effect  (a  P.D.), 
or  else  an  effect  (a  P.D.)  without  a  cause  (an  E.M.F.). 

76.  The  E.M.F.  of  a  galvanic  cell.    In  a  galvanic  cell  (Fig.  84) 
there  is  an  E.M.F.  at  every  surface  of  contact  of  two  dissimilar 
substances,  and  the  total  E.M.F.  of  the  cell  is  simply  the  algebraic 
sum  of  all  these  E.M.F.'s  of  contact.    It  is  measured  by  the  total 
P.D.  which  the  cell  is  able  to  maintain  between  its  terminals  A 


there  would,  of  course,  be  a  tendency  to  diffuse  more  rapidly  in  one  direction 
than  in  the  other,  and  this  fact  would  have  to  be  taken  into  account,  in 
connection  with  the  specific  attractions,  in  order  to  determine  which  of  two 
substances  would  have  an  excess  of  negative  electrons. 


114 


ELECTRICITY,  SOUND,  AND  LIGHT 


a 

0 


FIG.  84 


and  B  on  open  circuit.  That  is,  the  resultant  E.M.F.  of  contact 
causes  a  positive  charge  to  accumulate  on  A  and  a  negative  charge 
on  B  until  the  back  pressure  through  the 
cell,  set  up  by  the  accumulation  of  these 
charges,  just  balances  the  total  E.M.F. 
which  is  producing  it,  and  thus  brings 
to  rest  the  action  of  this  E.M.F. 

The  water  analogy  of  the  galvanic  cell 
is  presented  in  Figure  85.  The  weight  W 
corresponds  to  the  E.M.F.  of  the  cell.  It 
is  the  cause  of  the  movement  of  the  water 
from  B  to  A.  Furthermore,  just  as  the 
weight  W  is  brought  to  rest  as  soon  as 
the  back  force  against  the  paddle  wheels, 
due  to  the  difference  in  levels  in  A  and  B,  becomes  equal  to  the 
forward  force  which  W  is  able  to 
exert,  so  the  actions  at  the  sur- 
faces of  contact  of  the  dissimilar 
substances  in  a  galvanic  cell  are 
brought  to  rest  by  the  P.D.  which 
the  E.M.F.  of  the  cell  sets  up  be-, 
tween  its  terminals. 

77.  Internal  resistance  of  a 
galvanic  cell.  Connecting  the  ter- 
minals A  and  B  of  the  galvanic  cell 
by  a  conductor  (Fig.  86)  causes 
the  P.D.  between  them  at  once  to 
fall,  just  as  connecting  the  tanks 
A  and  B  by  a  pipe  mn  (Fig.  87) 
causes  the  difference  in  level  be- 
tween them  to  diminish.  Further, 
just  as  the  weight  W  at  once  be- 
gins to  cause  the  wheel  to  rotate 
and  thus  to  restore  the  lost  differ- 
ence in  level,  so  the  E.M.F.  of  the 
cell  begins  at  once  to  act  to  restore  the  lost  P.D.  between  the  ter- 
minals of  the  cell  as  soon  as  they  are  connected  by  a  conductor. 


FIG.  85 


ELECTROMOTIVE  FORCE 


115 


A 

ff! 


D 

e 


FlG. 


The  fact  that  a  current  flows  through  the  circuit  composed  of 
the  conductor  AB,  the  dissimilar  metals  Cu  and  Zn,  and  the  liquid 
in  the  cell,  means  that  there  is  a  fall  in 
potential  from  A  to  B  in  the  external,  and 
from  B  to  A  in  the  internal,  portion  of  the 
circuit.  The  total  fall  in  potential  through- 
out all  of  the  conductors  of  the  circuit 
(Fig.  86)  is  equal  to  the  resultant  E.M.F. 
of  contact  of  the  dissimilar  substances  in 
the  circuit,  just  as  in  the  case  of  Fig- 
ure 83  total  P.P.  equals  resultant  E.M.F. 
Similarly,  in  the  case  of  the  water  analogy 
of  Figure  87,  the  water  flows  through  the 
external  circuit  mn,  and  through  the  in- 
ternal circuit  po ;  and  the  sum  of  the  falls  in  pressure  in  the 

external  and  the  internal  portions 
of  the  circuit  is  equal  to  the  differ- 
ence in  pressure  produced  by  the 
wheel.  In  the  case  of  a  galvanic 
cell  this  E.M.F.  is  the  result  of  a 
difference  in  the  energy  of  combi- 
nation of  the  zinc  with  the  liquid 
(for  example,  sulphuric  acid)  and 
the  energy  of  combination  of  the 
copper  with  the  liquid.  In  the  case 
of  the  thermal  E.M.F.'s  the  result- 
ant E.M.F.  was  the  direct  result  of 
energy  supplied  to  one  junction  in 
the  form  of  heat. 

How  near  to  the  original  P.D. 
the  E.M.F.  of  the  cell  is  able  to 
maintain  the  terminals  A  and  B 
(Fig.  86)  when  they  are  connected 
by  the  conductor  depends  wholly 
upon  the  relative  amounts  of  diffi- 
culty which  the  conductor,  on  the  one  hand,  has  in  discharging 
A  and  B,  and  which  the  E.M.F.,  on  the  other  hand,  has  in  forcing 


FIG.  87 


116  ELECTRICITY,  SOUND,  AND  LIGHT 

new  charges  through  the  cell  up  to  these  terminals  ;  that  is,  it 
depends  upon  the  relative  resistances  of  the  internal  and  the 
external  portions  of  the  circuit  (see  p.  47).  Similarly,  the  differ- 
ence in  water  level  maintained  by  W  between  A  and  B  (Fig.  87) 
will  depend  wholly  upon  the  relative  capacities  of  the  pipes  mn 
and  op  for  carrying  water. 

If  the  resistance  of  the  conductor  connecting  A  and  B  (Fig.  86) 
is  very  large  in  comparison  with  the  resistance  of  the  liquid  be- 
tween Z  and  C,  then  the  P.D.  maintained  between  A  and  B  will 
be  practically  the  same  as  upon  open  circuit.  But  in  any  case  the 
total  E.M.F.  of  the  cell  must  be  equal  to  the  sums  of  the  falls  in 
potential  in  the  external  and  internal  portions.  Thus  if  I  denote 
the  current  furnished  by  the  cell,  R  its  external  and  r  its  internal 
resistance,  and  if  PD^  and  PD2  denote  external  and  internal  falls  in 
potential  respectively,  then,  by  Ohm's  law,  PD1  =  RI,  and  PD2  =  rl\ 
or,  since  PD1  +  PD2  =  EHF,  we  have 

EMF  =  I(R  +  r).  (1) 

Otherwise  stated,  while  Ohm's  law  applied  to  any  part  of  a  circuit 
is  PD/I  =  R,  as  applied  to  a  complete  circuit  it  is 


78.  Polarizing  and  nonpolarizing  cells.  A  cell  which  consists 
simply  of  two  dissimilar  metals,  for  example  zinc  and  copper, 
immersed  in  a  conducting  liquid  like  sulphuric  acid,  will,  in  gen- 
eral, have  neither  a  constant  E.M.F.  nor  a  constant  internal  resist- 
ance ;  for,  since  the  positive  ions  which  are  already  in  the  liquid 
are  of  hydrogen,*  as  soon  as  the  zinc  begins  to  go  into  solution  in 
the  form  of  positively  charged  zinc  ions  which  repel  the  positive 
hydrogen  ions  away  from  the  zinc  plate  and  toward  the  copper 
plate,  these  repelled  hydrogen  ions  begin  to  accumulate  about 
the  copper  plate  and  there  to  change  into  neutral  hydrogen  mole- 
cules, and  thus  to  alter  completely  the  character  of  the  surface  of 
the  plate.  In  effect  the  plate  of  copper  is  replaced  by  a  plate  of 

*  H2S04  +  H20  -*  H20  +  211  +  S04  (see  Chap.  XVI). 


ELECTEOMOTIYE  FOECE 


117 


hydrogen.  This  alters  completely  the  E.M.F.  of  contact  at  the 
surface  of  this  plate,  and  it  also  alters  the  internal  resistance  of 
the  cell.  A  cell  of  this  sort  is  said  to  be  a  polarizing  cell,  because 
the  current  which  it  furnishes  diminishes  rapidly  as  the  hydrogen 
accumulates. 

A  nonpolarizing  cell  is  one  in  which  the  plate  toward  which  the 
ions  are  urged  is  immersed  in  a  solution  of  a  salt  of  the  same  metal 
as  the  plate ;  for  example,  a  copper  plate  in  a  copper  sulphate 
solution.  The  character  of  the  surface  of  this  plate  is  then  quite 
unchanged  by  the  passage 
of  the  current,  for  the  sub- 
stance deposited  is  the  same 
as  the  substance  of  which 
the  plate  is  composed.  The 
most  familiar  of  the  nonpo- 
larizing cells  is  the  Daniell 
cell  (Fig.  88).  It  consists 
of  a  jar  partly  filled  with 
a  copper  sulphate  (CuS04) 
solution  in  which  is  placed 
a  porous  earthenware  cup 
containing  a  solution  of  zinc 
sulphate  (ZnS04).  In  the  zinc  sulphate  is  a  zinc  plate  and  in  the 
copper  sulphate  a  copper  plate.  The  porous  cup  is  simply  to  keep 
the  liquids  from  mixing  and  yet  to  permit  the  passage  of  the  ions, 
which  travel  with  relative  ease  through  its  pores.  Such  a  cell,  if 
freshly  set  up,  will  have  both  an  E.M.F.  and  an  internal  resistance 
which  are  very  nearly  constant.  Since  the  copper  is  continually 
passing  out  of  solution,  the  compartment  S  is  usually  kept  filled 
with  copper  sulphate  crystals  so  as  to  keep  the  solution  saturated. 

79.  Methods  of  measuring  the  E.M.F.  and  internal  resistances 
of  Daniell  cells.  One  of  the  simplest  and  most  satisfactory  ways 
of  measuring  the  E.M.F.  and  internal  resistance  of  a  Daniell  cell 
is  as  follows.  First  connect  with,  say,  No.  18  copper  wire,  the 
terminals  of  the  cell  directly  to  the  terminals  of  a  voltmeter  and 
read  the  P.D.  indicated.  Then  replace  the  voltmeter  by  an  am- 
meter and  read  the  current  furnished.  If  Rv  and  It  ±  represent 


FIG. 


118  ELECTRICITY,  SOUND,  AND  LIGHT 

the  resistances  of  the  voltmeter  and  ammeter  respectively  (quan- 
tities which  may  be  assumed  to  be  known),  PD  the  voltmeter 
reading,  and  /  the  ammeter  reading,  then,  by  Ohm's  law,  the  cur- 
rent sent  through  the  voltmeter  in  the  first  of  the  above  obser- 
vations is  PD/RV.  But  this  same  current  is  also  equal  to  — 
Hence  our  first  equation  connecting  EMF  and  r  is  F 

EMF_(Rv+r) 
PD   =         Rv 

The  equation  which  corresponds  to  the  second  observation  is 

<^)=£  (4) 

We  have  only  to  solve  (3)  and  (4)  in  order  to  obtain  both  EMF  and  r. 
When  r  is  small  as  compared  to  Rv,  equation  (3)  gives  EMF=PD 

(the  voltmeter  reading)  as  a  first  approximation.  Now  for  the 

determination  of  a  quantity  so  little  con- 
stant as  r  it  will,  in  general,  be  suffi- 
ciently accurate  to  make  the  assumption 
that  the  E.M.F.  of  the  cell  is  the  volt- 
meter reading,  and  then  to  solve  equa- 
tion (4)  for  r.  The  value  of  the  E.M.F. 
correct  to  a  second  approximation,  which 
will  generally  be  as  accurate  as  the  volt- 
meter can  be  read,  may  then  be  obtained 
by  substituting  the  value  of  r  as  thus  found 

"Fir    89 

in  equation  (3),  and  solving  for  EMF. 

A  second  method  of  measuring  internal  resistance,  and  one 
which  has  the  advantage  of  making  it  possible  to  determine 
whether  or  not,  with  the  cell  employed,  r  varies  largely  with 
the  current  used,  is  the  following.  The  cell  B,  a  voltmeter  V,  a 
resistance  box  R,  and  a  key  K  are  connected  as  in  Figure  89, 
the  resistances  of  contacts  and  connections  between  K  and  R 
being  made  as  small  as  possible.  The  voltmeter  reading  when 
the  key  is  up  is  approximately  the  E.M.F.  of  the  cell.  The 
voltmeter  reading  when  the  key  is  down  is  the  P.D.  across  the 


INTERNAL  RESISTANCE  119 

resistance  R.  If  R  is  small  in  comparison  with  the  resistance  of 
the  voltmeter,  then  the  current  furnished  by  the  cell  when  the 
key  is  down  is  the  reading  of  the  voltmeter  (which  we  shall  call 
simply  PD)  divided  by  R.  But  this  same  current  is  also  given 
by  dividing  the  E.M.F.  of  the  cell  by  R  +  r.  Hence  we  have 

EMF       PD 


(R  +  r)        R 

R(EHF-PD) 


In  the  use  of  this  method,  if  results  correct  to  even  so  much  as  5  per 
cent,  for  example,  are  to  be  obtained,  R  should  never  be  made  more 
than  a  twentieth  of  the  resistance  of  the  voltmeter.  Furthermore, 
instead  of  using  the  voltmeter  reading  when  the  key  is  open  as 
the  E.M.F.  of  the  cell,  it  is,  of  course,  more  accurate  to  raise  this 
reading  to  the  true  E.M.F.  by  substituting  in  equation  (3)  the 
approximate  value  of  r,  known  by  a  rough  preliminary  observation 
either  by  this  method  or  its  predecessor. 

EXPERIMENT  10 

Object.  To  determine  the  E.M.F.  and  the  internal  resistance  of  a  Daniell 
cell. 

Directions.  I.  Clean  the  zinc  of  the  Daniell  cell  carefully  so  as  to  remove 
any  deposit  of  copper  or  copper  oxide  which  may  have  accumulated  because 
of  the  diffusion  of  the  copper  sulphate  through  the  porous  cup,  and  which 
ivS  a  fruitful  source  of  polarization  in  the  cell.  Clean  also  the  porous  cup 
and  put  in  a  fresh  solution  of  zinc  sulphate. 

II.  Determine  the  resistance  of  the  coarser  register  of  the  milliammeter 
either  by  the  Wheatstone-bridge  method  or  else  by  sending  a  current  from 
the  cell  through  it  and  the  voltmeter  arranged  in  parallel.    In  the  latter 
case  the  reading  of  the  voltmeter  divided  by  the  reading  of  the  ammeter 
will  be  the  resistance  of  the  latter.    If  the  current  is  too  large  for  the 
ammeter,  reduce  it  by  inserting  a  few  feet  of  German  silver  wire. 

III.  Determine  the  resistance  of  the  voltmeter  either  by  the  bridge 
method,  or  else   by  putting  the  milliammeter  (low  register)  and  the  volt- 
meter in  series  in  the  circuit  of  the  cell.    The  reading  of  the  voltmeter 
divided  by  the  reading  of  the  milliammeter  will  be  the  resistance  of  the 
voltmeter. 


120  ELECTRICITY,  SOUND,  AND  LIGHT 

IV.  Connect  the  high  register  of  the  milliammeter  directly  to  the  ter- 
minals of  the  cell,  or  insert,  if  necessary  to  keep  the  deflection  on  the  scale, 
a  small  length  of  German  silver  wire  of  known  resistance  per  meter  and 
read  the  current ;  then  replace  the  ammeter  by  the  voltmeter  and  read  the 
P.D.    Compute  r  and  EMF  as  indicated  in  section  79,  equations  (3)  and 
(4),  subtracting,  of  course,  the  resistance  of  the  German  silver  wire  if  it 
is  used. 

V.  Connect  the  cell,  a  resistance  box,  a  key,  and  a  voltmeter,  as  indicated 
in  the  second  method  of  section  79,  using  a  short,  thick  wire  (e.g.  No.  16) 
to  connect  the  key  to  the  resistance  box,  and  seeing  to  it  that  the  contacts 
at  all  binding  posts  and  plugs  are  thoroughly  good. 

Compute  r  from  equation  (6)  for  at  least  three  different  values  of  R, 
e.g.  1,  2,  and  5  ohms,  and  decide  whether  or  not  the  internal  resistance  of 
a  Daniell  cell  varies  appreciably  with  the  current  taken  from  it. 


EXAMPLE 

I.  The  ammeter  reading  was  .382  ampere  when  the  reading  of  the  volt- 
meter shunted  across  it  was  .04  volt,  hence  RA  =  .10  ohm. 

II.  A  current  of  .0027  ampere,  as  measured  by  the  low  register  of  the 
milliammeter,  was  passed  through  the  voltmeter,  which  read  1.04  volts. 
Hence  Rv=  385  ohms. 

III.  The  voltmeter  connected  directly  to  the  Daniell  cell  used  read  1.05 
volts.    The  ammeter  when  connected  directly  read  .427  ampere.    Hence 
r  =  2.35  ohms,  and  EMF=  1.065  volts. 

IV.  A  P.D.  of  .33  volt  was  observed  across  the  terminals  of  a  resist- 
ance of  1  ohm.    Substitution  of  these  values  and  the  value  of  the  E.M.F. 
found  above  gave 

1.065  -  .33       0  00    , 
r  = =  2.23  ohms. 


Similarly  f or  R  =  2  ohms,  PD  =  .85  volt  and  r  =  2.39  ohms.     Also  for 
R  =  3  ohms,  PD  =  .60  volt  and  r  =  2.32  ohms. 


CHAPTEE  XI 
THE  COMPARISON  OF  ELECTROMOTIVE  FORCES 

80.  E.M.F.'s  of  polarizing  cells.    The  methods  described  in  the 
preceding  chapter  can  obviously  be  employed  only  with  cells,  like 
the  Daniell.  which  do  not  polarize.    But  having  once  found  the 
E.M.F.  of  any  nonpolarizing  cell,  the  E.M.F.'s  of  any  other  cells, 
polarizing  or  nonpolarizing,  can  be  very  easily  and  very  accurately 
determined  by  the  use  of  a  comparison  method  in  which  the  first 
cell  is  taken  as  a  standard.*    We  shall  consider  two  such  methods. 

81.  The  ballistic  galvanometer  method.    Since  the  E.M.F.  of  a 
cell  is  the  P.D.  which  it  maintains  between  its  terminals  on  open 
circuit,  it  is  only  necessary  to  connect  the  terminals  of  any  cell  to 
the  plates  of  a  condenser  in  order  to  charge  these  plates  to  a  P.D. 


*The  cell  most 
commonly  used  as  a 
standard  for  the  com- 
parison of  E.M.F.'s 
is  some  modification 
of  the  Clark  cell 
shown  in  section  in 
Figure  90.  When 
made  under  certain 
standard  conditions 
it  has  an  E.M.F. 
expressed  by  the 
following  relation 
where  t  is  the  tem- 
perature. 

E  =  1.434  x 

[1-. 00077(^-15°)]. 


Platinum  Wire 
Zn.  Amalgam 


Zn  S0,-m 


S04  Solution 


ZnSO, 

=»  Paste  ofHg2  SO, 

•--g» — -Platinum  Wire 
Pure  Hg. 


FIG.  90 


The  cell,  however,  is 
incapable  of  sup- 
plying an  appreciable  current  without  considerable  polarization.     It  is  there- 
fore serviceable  for  accurate  work  only  if  kept  on  open  circuit.    It'  will  be  seen 
that  it  may  be  used  according  to  either  of  the  two  methods  of  this  chapter. 

121 


122 


ELECTRICITY,  SOUND,  AND  LIGHT 


equal  to  the  E.M.F.  of  the  cell.  Since,  further,  the  quantity  of 
charge  on  the  plates  of  a  given  condenser  is  proportional  to  the 
P.D.  between  them  (Q  =  Cx  PD),  we  have  only  to  charge  the 
same  condenser  successively  by  means  of  different  cells  in  order  to 
obtain  charges  which  are  strictly  proportional  to  the  E.M.F.'s  of 
the  cells.  Since,  finally,  the  throws  of  a  given  ballistic  galvanom- 
eter are  proportional  to  the  charges  sent  through  it  (Q  = —  > 

see  p.  96),  we  have,  if  Ol  and  02  are  the  respective  throws  pro- 
duced by  discharging  a  given  condenser  through  a  ballistic  gal- 
vanometer when  the  condenser  has  been  charged,  first  by  a  cell  of 
E.M.F.  E19  and  second  by  a  cell  of  E.M.F.  Ev 


(1) 


82.  The  potentiometer  method.    The  only  disadvantage  of  the 
preceding  method  is  that    it  is  a  deflection    rather  than  a  zero 

method.    The  fol- 

i     i 

lowing  method  is 
just  as  faultless 
theoretically,  and 
it  can  be  made  ac- 
curate to  the  fifth 
or  sixth  place  of 
decimals  if  tem- 
perature and  other 
conditions  can  be 
kept  constant 
enough  to  make 
such  a  degree  of  accuracy  desirable.  A  storage  battery  B,  or 
any  combination  of  batteries  having  an  E.M.F.  which  is  con- 
stant and  somewhat  higher  than  that  of  any  of  the  cells  to  be 
compared,  is  connected  as  in  Figure  91  to  the  ends  of  a  wire  ab 
of  sufficiently  high  resistance  to  prevent  a  current  from  flowing 
which  is  large  enough  to  heat  the  wire  appreciably.  The  resist- 
ance of  ab  must  also  be  so  high  in  comparison  with  the  internal 
resistance  of  B  that  the  P.D.  maintained  by  B  between  a  and 


FIG.  91 


COMPAKISON  OF  ELECTEOMOTIVE  FORCES      123 

I  is  greater  than  the  E.M.F.  of  either  E^  or  Ev  the  two  cells  to 
be  compared.  These  cells  are  connected  as  in  the  figure,  so  that 
their  negative  terminals  are  joined  to  the  same  point  a  to  which 
the  negative  terminal  of  B  is  connected,  their  positive  terminals 
being  connected  to  the  contact  points  m  and  n  of  a  double  switch 
S,  through  which  either  cell  can  be  put  into  connection  with 
a  resistance  box  R,  a  galvanometer  Gy  and  a  wire  w  which  can 
be  touched  at  any  point  along  the  wire  ab.  The  comparison  of 
the  E.M.F.'s  is  made  as  follows.  Suppose  that  the  switch  S  is 
turned  so  as  to  touch  the  contact  m  and  thus  put  the  cell  El 
into  the  galvanometer  circuit.  If  now  the  free  terminal  of  the 
wire  w  were  to  be  touched  to  any  point  on  ab,  for  example  c,  a 
current  would  always  flow  through  G  and  R  from  right  to  left 
provided  there  were  no  cell  in  the  circuit  cGEma.  The  cell  E1 
which  is  in  this  circuit,  however,  tends  to  force  current  in  the 
opposite  direction,  namely  from  left  to  right  through  R  and  G. 
If,  then,  the  P.D.  which  already  exists  between  c  and  a,  when  the 
wire  is  touched  at  c,  is  greater  than  the  E.M.F.  Ev  a  current  will 
actually  flow  through  G  and  R  from  right  to  left;  but  if  the  E.M.F. 
of  the  cell  El  is  greater  than  the  P.D.  which  is  maintained  between 
the  points  c  and  a  by  the  battery  B,  then  a  current  will  flow 
through  the  galvanometer  from  left  to  right.  If  the  P.D.  between 
c  and  a  is  exactly  equal  to  the  E.M.F.  Ev  then  no  current  what- 
ever will  flow  through  the  circuit  of  the  cell,  that  is,  through  G. 
We  have  then  only  to  find  the  point  on  ab  which  can  be  touched 
by  the  free  end  of  the  wire  without  producing  any  galvanometer 
deflection  whatever,  in  order  to  obtain  the  point  such  that  the  P.D. 
between  it  and  a  is  exactly  equal  to  the  E.M.F.  of  the  cell. 

Suppose  now  that  the  switch  S  is  turned  so  as  to  make  contact 
with  the  terminal  n  of  the  other  cell.  If  it  is  now  found  that 
some  other  point  d  is  the  point  for  which  the  galvanometer  shows 
no  deflection,  then,  if  the  wire  ab  is  uniform,  we  have 

El  _  PD  from  c  to  a  _  length  ac 
E^      PD  from  d  to  a      length  ad 

While  the  point  of  no  deflection  is  being  found,  the  resist- 
ance R  should  be  made  very  large  (e.g.  20,000  ohms),  for  then  no 


124 


ELECTRICITY,  SOUND,  AND  LIGHT 


appreciable  current  will  flow  through  the  cell  circuit,  and  hence 
this  cell  will  not  polarize,  even  if  it  be  one  of  the  polarizing  kind. 
After  the  point  of  zero  deflection  is  found,  the  resistance  R  may 
be  varied,  or  in  fact  entirely  removed,  without  altering  the  point 
of  balance,  for  obviously  at  this  point  the  cell  is  in  exact  equilib- 
rium with  the  P.D.  between  c  and  a;  that  is,  it  is  virtually  on  open 
circuit.  The  only  reason  for  introducing  R  at  all  was  to  prevent 
the  cell  from  polarizing  while  the  point  of  balance  was  being 
found,  and  to  protect  the  galvanometer  from  too  violent  deflec- 
tions. Varying  the  value  of  R  will  then  alter  nothing  save  the 
sharpness  with  which  the  point  of  zero  deflection  can  be  located. 


EXPERIMENT  11 

(A)  Object.  To  find  the  E.M.F.  of  a  Leclanch6*  cell  by  comparing  it 
with  a  Daniell  cell  by  the  ballistic-galvanometer  method. 

Directions.  First  determine  the  E.M.F.  of  the  Daniell  cell  El  by  the 
method  of  Experiment  10.  Then  set  up  Ev  the  Leclanche"  cell  E2,  the 
double  switch  5,  the  condenser  C,  the  discharge  key  K,  the  galvanom- 
eter G,  and  the  damping  key,  in  the  manner  shown  in  the  diagram  (Fig.  93). 
Be  very  careful  in  so  doing  not  at  any  time  to  short  circuit  the  Leclanche" 
cell,  for  if  it  is  once  allowed  to  become  polarized,  it  may  take  it  hours  to 
recover  entirely  its  original  E.M.F.  Take  in  succession  the  throws  of  each 
cell.  From  the  known  E.M.F.  of  the  Daniell  cell  and  the  mean  of  at  least 
six  throws,  compute  the  E.M.F.  of  the  Leclanch6  cell. 

*  The  Leclanch4  cell,  shown  in  Figure  92,  is  of  wide  commercial  use  for  cir- 
cuits closed  only  intermittently.  It  consists  of  zinc  and  carbon  electrodes  in 
a  solution  of  ammonium  chloride  (NH4C1).  The  carbon  electrode  is  commonly 
in  a  porous  cup  filled  with  manganese  dioxide  (Mn()2). 
The  action  of  the  cell  is  as  follows.  The  zinc  goes  into 
solution  in  the  ammonium  chloride  in  the  form  of  posi- 
tively charged  zinc  ions.  This  gives  a  positive  charge 
to  the  solution  about  the  zinc  plate,  and  hence  the  posi- 
tive ions  already  in  solution  (NH4)  are  driven  toward 
the  carbon  plate.  Instead  of  going  out  of  solution,  how- 
ever, they  decompose  the  water  (H2O)  about  this  plate 
and  form  ammonium  hydroxide  (NII4OH)  and  free  II 
ions,  which  are  driven  out  of  solution  at  the  plate  in 
the  form  of  hydrogen  gas.  The  function  of  the  MnO2  is 
to  dispose  of  this  hydrogen  by  uniting  with  it  to  form 
Mn2O3  (manganic  oxide)  and  water.  This  depolarizing 

action  of  the  MnO2  is  quite  slow,  however,  so  that  the  cell  polarizes  rapidly  on 
short  circuit,  but  recovers  completely  when  left  to  itself  for  a  few  hours. 


FIG.  92 


COMPARISON  OF  ELECTROMOTIVE  FORCES      125 

(B)  Object.  To  compare  the  E.M.F. 's  of  the  Daniell  and  Leclanche' 
cells  by  the  potentiometer  method. 

Directions.  Connect  as  in  Figure  91,  being  very  careful  not  at  any  time 
to  short  circuit  the  Leclanche  cell.  Let  II  be  a  resistance  of  from  10,000 


FIG.  93 


to  100,000  ohms  while  the  point  of  balance  is  being  found.  If  the  point 
of  no  deflection  is  not  sharply  marked,  reduce  R  after  the  point  of  balance 
has  been  approximately  located.  Compare  results  with  those  obtained 
in  (A). 

EXAMPLE 

(A)  The  voltmeter  reading  for  a  Daniell  cell  was  1.080  volts.    The  volt- 
meter resistance  was  197.5  ohms  and  the  internal  resistance  of  the  cell  as 
found  by  the  first  method  of  section  79  was  2.48  ohms ;  hence  the  E.M.F. 
of  the  Daniell  cell  was  1.094  volts.     A  condenser  of  .1 -microfarad  capacity 
charged  by  this  E.M.F.  gave  a  throw  of  3.00  cm.  on  the  galvanometer  used. 

The  same  condenser  when  charged  by  the  Leclanche  cell  caused  a  throw 

4  os 
of  4.28  cm.   Hence  the  E.M.F.  of  the  Leclanch£  cell  = x  1.094  =  1.563 

volts.  :W)0 

(B)  For  the  Daniell  cell  a  balance  was  obtained  when  ac  was  50.06  cm., 
for  the  Leclanche"  cell  when  ad  was  71.10  cm.    Hence  the  E.M.F.  of  the 

Leclanche'  cell  =  *    '       x  1.094  =  1.556   volts.     The  values  found  by  the 

50.06 
two  methods  differed  by  .45  of  one  per  cent. 


CHAPTER  XII 
ELECTRO-MAGNETIC  INDUCTION 

83.  The  principle  of  electro-magnetic  induction.    Thus  far  our 

discussion  of  electrical  phenomena  has  been  centered  about  Oersted's 
discovery  that  an  electric  current  is  surrounded  by  a  magnetic  held. 
We  now  come  to  the  discussion  of  the  applications  of  an  exceed- 
ingly important  discovery,  published  for  the  first  time  by  Faraday 
in  1831,  although  the  discovery  itself  was  doubtless  made  a  year 
earlier  by  Joseph  Henry,  at  that  time  a  high-school  teacher  in 
Albany.  It  is  to  the  discovery  of  the  principle  of  electro-magnetic 
induction  that  we  owe  the  modern  dynamo,  the  induction  coil,  the 
transformer,  the  telephone,  and  most  of  the  other  practical  appli- 
cations of  electricity  to-day.  This  principle  may  be  stated  as  fol- 
lows :  Whenever  a  conductor  moves  in  a  magnetic  field  in  such  a 
way  as  to  cut  lines  of  magnetic  force,  an  E.M.F.  is  induced  in  it. 
Thus  when  the  vertical  wire  ac,  shown  in  Figure  94,  is  moved 
across  the  field,  an  E.M.F.  is  induced  in  it,  and  since,  in  this  case, 
the  circuit  is  completed  by  the  wire  abc,  this  E.M.F.  causes  a  cur- 
rent to  flow  about  the  circuit.  No  current  whatever  is  found  to 
be  induced  when  the  conductor  moves  parallel  to  the  lines  of 
force.  These  statements  can  be  easily  verified  by  moving  a  single 
wire,  to  the  ends  of  which  a  sensitive  galvanometer  is  attached, 
first  in  a  direction  perpendicular  to,  then  in  a  direction  parallel 
with,  the  field  of  a  strong  magnet. 

84.  Direction  of  the  induced  E.M.F.    Since  energy  is  expended 
when  an  electrical  current  flows  in  a  wire,  it  follows  from  the 
principle  of  the  conservation  of  energy  that  work  must  be  done  in 
inducing  this  current,    Now  we  have  already  seen  in  sections  50 
and  51  (p.  78)  that  any  current  in  ac  (Fig.  94)  interacts  with  the 
magnetic  field  in  such  a  way  as  to  tend  to  push  the  wire  across  the 
field.    If,  then,  work  must  be  done  in  inducing  the  current  in  ac, 

126 


ELECTRO-MAGNETIC  INDUCTION 


127 


it  follows  that  the  direction  of  the  induced  current  must  be  such 
that  the  mechanical  force  with  which  it  is  urged  across  the  field 
constitutes  the  reaction  against  which  the  work  is  done ;  i.e.  this 
mechanical  force  must  be  opposite  to  the  direction  of  motion. 
For  if  the  induced  current  could  be  in  the  opposite  direction,  the 
wire  would  be  urged  across  the  field  in  the  direction  in  which  it 
is  going  and  electrical  energy  would  be  created  indefinitely  with- 
out any  expenditure  of  mechanical  energy  whatever.  The  induced 


Field 


FIG.  94 


current  must  then  be  in  a  direction  such  as  to  call  into  play  a  force 
which  opposes  the  motion  inducing  it.  This  statement,  which  is 
seen  to  follow  from  the  principle  of  the  conservation  of  energy,  is 
known  as  Lenz's  law. 

A  current,  of  course,  cannot  flow  unless  the  circuit  is  complete, 
but  an  E.M.F.  which  would  cause  a  current  to  flow  in  a  direction 
such  as  to  oppose  the  motion  inducing  it,  is  nevertheless  always 
induced  when  a  conductor  cuts  lines  of  force,  and  this  E.M.F. 
causes  positive  and  negative  charges  to  appear  upon  the  ends  of 


128  ELECTKICITY,  SOUND,  AND  LIGHT 

the  open  circuit  just  as  the  E.M.F.  of  a  cell  causes  such  charges 
to  appear  upon  the  terminals  of  the  cell. 

It  is  possible  to  formulate  for  induced  currents  a  rule  connect- 
ing the  directions  of  the  motion,  of  the  magnetic  field,  and  of  the 
current,  similar  in  form  to  the  motor  rule,  but  making  use  of  the 
right  hand  instead  of  the  left.  Thus  if  the  thumb  and  the  first 
two  fingers  of  the  right  hand  are  extended  in  directions  at  right 
angles  to  one  another,  and  if  the  forefinger  is  made  to  point  in  the 
direction  of  the  magnetic  lines  and  the  thumb  in  the  direction  of 
the  component  of  the  motion  of  the  conductor  which  is  at  right 
angles  to  these  lines,  then  the  second  finger  will  point  in  the  direc- 
tion of  the  induced  current.  It  will  be  obvious  from  a  consideration 
of  Figure  94  that  this  rule  is  only  another  form  of  statement  of 
Lenz's  law.  From  its  especial  application  to  the  dynamo  it  is 
known  as  the  dynamo  rule. 

85.  The  value  of  an  induced  E.M.F.  The  quantitative  expres- 
sion for  an  induced  E.M.F.  may  be  found  by  a  consideration  of 
Figure  95.  A  and  B  represent  heavy  conducting  bars  of  negli- 
gible resistance,  between 
f —  the  terminals  of  which 


.    L    .    .    .  is  connected  the  coil  D. 

•    !••••  The  line  ab  represents  a 

sliding  bar,  also  of  negli- 


gible   resistance.     The 
dots  represent  the  cross 
i  section  of  a  uniform  rnag- 

i  ,  * —     netic  field  of  intensity 


d 

F      95  ofC  perpendicular  to  the 

plane    of    the   paper,    in 

which  this  metal  framework  is  supposed  to  lie.  A  force  F  acts 
on  the  slider  ab,  moving  it  with  uniform  speed  through  a  small 
distance  dx  to  a  new  position  a'bf.  This  motion  induces  an 
E.M.F.  which  causes  to  flow  around  the  circuit  a'Db'  a  current 
of  /  absolute  units.  The  reaction  of  the  field  &C  on  the  /  cm.  of 
length  of  the  slider  is  a  force  equal' to  F  and  of  value  II 3i, 
dynes  (sect.  50,  p.  78).  The  mechanical  work  done  in  moving 
the  coil  dx  cm.  against  this  reaction  is  Ildidx  ergs.  If  E 


ELECTKO-MAGNETIC   INDUCTION  129 

represents  in  absolute  units  the  E.M.F.  causing  this  current  to 
flow,  and  if  dt  is  the  time  during  which  the  motion  is  taking 
place  (i.e.  the  time  during  which  we  are  considering  the  flow 
of  current),  then  the  electrical  energy  expended  by  the  current 
in  this  time  is  Eldt  ergs  (see  Chap.  IV).  Now  since  no  part 
of  the  original  mechanical  energy  is  transformed  into  forms 
other  than  electrical  (e.g.  chemical),  these  two  expressions,  the 
one  for  the  work  done  and  the  other  for  the  electrical  energy 
developed,  must  be  equal.  That  is, 


II  &djs  =  Eldt,  (1) 

IcKdx 
E=—.  (2) 

Now  since  Idx  represents  the  area  across  which  the  slider  has 
moved,  and  since  cK  represents  the  number  of  lines  per  square 
centimeter,  it  is  clear  that  Icftdx  is  the  total  number  of  magnetic 
lines  cut  in  the  time  dt,  and  hence  that  an  induced  E.M.F.  is 
numerically  equal  to  the  rate  at  which  magnetic  lines  of  force 
are  cut.  That  is,  one  absolute  electro-magnetic  unit  of  E.M.F. 
is  induced  in  a  conductor  which  cuts  one  magnetic  line  of  force 
per  second.  Since  a  volt  is  108.  absolute  units,  a  volt  is  induced 
in  a  conductor  when  it  is  cutting  lines  at  the  rate  of  100,000,000 
per  second. 

The  total  number  of  magnetic  lines  which  pass  through  any 
area  is  known  as  the  magnetic  flux  through  that  area,  or  simply 
as  the  flux.  Algebraically  stated,  if  <I>  represents  flux,  eft  field 
strength,  and  A  area,  then,  by  definition,  in  case  the  direction 
of  the  field  is  normal  to  the  area, 

$>  =  A$C.  (3) 

In  case  the  field  makes  an  angle  6  with  the  normal  to  the  area, 

then 

<£  =  AW  cos  0.  (4) 

That  is,  the  flux  across  any  area  is  the  product  of  the  field 
strength  by  the  projection  of  this  area  upon  a  plane  normal  to 


130  ELECTRICITY,  SOUND,  AKD  LIGHT 

the  direction  of  the  field.  In  the  expression  for  E  just  found, 
loKdx  evidently  represents  the  small  increment  of  flux  added 
(algebraically)  during  the  element  of  time  dt  to  the  flux  already 
inclosed  by  the  circuit.  Denoting  the  increment  of  flux  by  d<&, 
we  have 


That  is,  the  E.M.F.  in  absolute  units  induced  in  any  closed  circuit 
is  the  rate  of  change  of  the  included  flux.  To  be  adaptable  to  this 
expression,  Lenz's  law  may  be  restated  as  follows  :  The  E.M.F. 
induced  in  any  closed  circuit  will  cause  a  current  to  flow  in  a 
direction  such  as  to  oppose  the  change  in  flux  inducing  it. 

86.  General  statement  of  Lenz's  law.  As  was  seen  above,  the 
one  and  sufficient  condition  necessary  for  the  electro-magnetic 
induction  of  an  E.M.F.  in  a  conductor  is  that  the  conductor  shall 
cut  magnetic  lines  of  force.  This  cutting  may  take  place,  how- 
ever, in  a  variety  of  ways,  and  it  is  of  some  importance  to  deduce 


FIG.  96 

a  statement  for  Lenz's  law  which  will  be  applicable  to  all  cases. 
Thus  if  instead  of  moving  a  wire  through  a  magnetic  field,  as  in 
the  above  illustration,  we  push  a  magnet  N/S  from  left  to  right  up 
to  a  coil  B,  as  in  Figure  96,  some  of  the  lines  of  force  which  exist 
in  the  space  surrounding  the  magnet,  and  move  with  it,  are  cut 
by  the  coil,  and  consequently  a  current  flows  in  it  in  such  a  direc- 
tion that  the  field  it  sets  up  about  the  coil  opposes  the  approach 
of  the  magnet ;  for  this  case  is  obviously  merely  the  converse  of 
that  considered  in  the  preceding  paragraph.  Furthermore,  pulling 
the  coil  away  from  the  magnet  must  cause  a  cutting  of  lines  in 


ELECTRO-MAGNETIC  INDUCTION 


131 


the  opposite  direction,  and  hence  a  current  in  such  a  direction  as 
to  attract  the  receding  magnet ;  that  is,  to  oppose  the  separation. 

Again,  if  instead  of  pushing  up  the  magnet  NS  we  set  near  B 
a  second  coil  A  (Fig.  97),  which  is  connected  to  the  circuit  of  a 
battery  D  through  a  key  K,  and  then  close  K,  we  obviously  cause 
B  to  be  cut  by  the  lines  of  force  which  spring  into  existence 
about  A,  precisely  as  though  we  had  thrust  NS  up  to  B  as  above. 
The  induced  current  in  B  must  then  be  in  such  a  direction  as  to 
cause  repulsion  between  A  and  B\  that  is,  it  must  thrust  lines  of 
force  through  B  in  a  direction  opposite  to  that  in  which  the  rising 
current  in  A  is  thrusting  lines  through  B.  Conversely,  when  K 
is  opened,  the  current  induced  in  B  must  be  in  such  a  direction  as 
to  cause  attraction  between  A  and  B]  that  is,  in  such  a  direction 


\     I 
\     ( 


^  '  /  itJ-^L. 

'  *  •/  /XZIZIZ.Z 


,  FIG.  9.7 

as  to  thrust  lines  through  B  in  the  same  direction  as  that  of  the 
lines  which  are  disappearing  from  B  because  of  the  decay  of  the 
current  in  A.  In  its  most  general  form  Lenz's  law  may  now  be 
stated  thus  :  Whenever  an  E.M.F.  is  induced  by  the  relative  motion 
of  a  conductor  and  magnetic  lines,  the  induced  current  will  always 
floiv  in  such  a  direction  as  to  oppose  the  change  which  is  induc- 
ing it.  If  the  change  is  a  mechanical  one,  the  induced  current  is  in 
such  a  direction  as  to  oppose  the  motion ;  if  the  change  is  a  mag- 
netic one,  the  induced  current  is  in  such  a  direction  as  to  oppose 
the  magnetic  change  which  is  taking  place. 

87.  The  ideal  dynamo.  The  most  practical  application  of  this 
principle  of  electro-magnetic  induction  is  to  the  dynamo,  a  machine 
for  converting  mechanical  energy  into  electrical  energy.  In  its 
simplest  form  it  consists  of  a  rectangular  coil  of  wire,  rotating 


132 


ELECTRICITY,  SOUND,  AND  LIGHT 


about  the  axis  of  its  length  in  a  uniform  magnetic  field,  the  direc- 
tion of  which  *is  perpendicular  to  its  axis.  Suppose  the  coil  to 
consist  of  one  turn  of  wire  and  to  be  represented  in  cross  section 
by  the  points  A  and  B  of  Figure  98.  The  field  of  strength  <?f,  in 
which  it  is  free  to  rotate,  is  represented  by  a  few  of  its  lines. 
Suppose  the  coil  to  be  rotated  from  a  position  A'B'  to  that  of  A!!B" 
through  a  small  angle  2  a.  Let  its  mean  angular  displacement 
from  its  original  position  at  right  angles  to  the  field  be  denoted 


FIG.  98 

by  0.  The  number  of  lines  cut  by  the  wire  A  is  proportional  to 
A'D,  and  if  I  represents  the  length  of  the  coil,  it  is  equal  to 
l&C  (A'D).  This  expression  divided  by  the  time,  dt  seconds,  dur- 
ing which  the  motion  took  place,  gives  the  average  value  of  the 
E.M.F.  induced  in  the  wire  A  when  displaced  by  an  angle  6 
from  its  original  position.  Now,  since  the  wire  B  cuts  an  equal 
number  of  lines  in  the  same  time,  it  has  induced  in  it  an  equal 
E.M.F.  But  this  E.M.F.  is  on  the  opposite  side  of  the  loop  and 
also  in  the  opposite  direction ;  hence  it  tends  to  cause  a  current 


ELECTRO-MAGNETIC  INDUCTION  133 

about  the  loop  in  series  with  that  in  A  ;  i.e.  the  total  E.M.F.  Ee  in- 
duced in  the  given  position  is  twice  the  amount  given  above.   That  is, 


This  may  be  put  in  a  different  form  by  the  following  substitu- 
tions. Since  the  angle  2  a  subtended  by  the  chord  A'  A"  is  small, 
the  chord  may  be  written  for  the  corresponding  arc.  Therefore 
A'D  =  A'  A"  x  sin  A'A"D.  But  angle  A'A"D  =  6.  Therefore,  upon 
substitution,  2  A'  A"  Iff  sin  0 

*"  -  dt  --  (7) 

Now  A'A",  the  arc  or  chord,  divided  by  dt,  is  the  linear  speed  of  the 
moving  wire.  Representing  the  radius  of  the  coil  by  r  and  the  uniform. 
angular  velocity  by  <w,  since  linear  speed  equals  angular  speed  times 
the  radius,  we  have 


This  gives  Ee=2rl  JCco  sin  d.  (9) 

Now  2  rl  is  the  area  of  the  coil,  and  when  multiplied  by  cK  it  is 
the  maximum  flux  through  it,  —  that  is,  the  flux  through  the  coil 
when  in  its  original  position  perpendicular  to  the  field.  Repre- 
senting this  flux  by  O,  we  get  from  equation  (9) 

E9  =  3>o)  sin  d.  (10) 

If  the  coil  consists  of  N  loops  in  series  instead  of  a  single  loop, 
the  same  E.M.F.  is  induced  in  each,  and  these  separate  E.M.F.'s 
must  be  added  to  get  the  total  value  of  EQ.  That  is, 

Ee  =  N&a>Bm0.  (11) 

This  is  the  E.M.F.  in  absolute  units.    Expressed  in  volts  it  is 

Ee  =  N<&cosm0W-*.  (12) 

Since,  for  any  given  case,  N  and  <E>  are  constants,  if  o>,  the 
angular  velocity,  is  kept  constant,  the  induced  E.M.F.  varies  directly 
as  the  sine  of  the  angular  displacement  of  the  coil  from  a  position 
at  right  angles  to  the  field.  A  method  for  showing  this  relation 
will  now  be  described. 


134 


ELECTRICITY,  SOUND,  AND  LIGHT 


88.  Experimental  analysis  of  dynamo  induction.  The  ideal 
dynamo  shown  in  Figure  99  is  arranged  so  that  at  every  revolu- 
tion of  the  crank  the  ratchet  is  lifted,  allowing  a  spring  to  rotate 
the  coil  through  10°.  The  terminals  of  the  coil  are  connected 
to  two  separately  insulated  copper  rings  attached  to  the  shaft. 
Copper  strips  or  "brushes"  bearing  upon  this  ring  are  connected 
to  a  moving-coil  galvanometer.  As  the  coil  moves  through  any 
ten-degree  interval,  an  E.M.F.  is  induced  which  corresponds  to  the 
mean  angular  displacement  of  the  coil  from  its  original  position. 
Since  the  total  resistance  through  which  this  E.M.F.  causes  a  cur- 
rent to  flow  remains  the  same  throughout  the  experiment,  the 
quantity  of  electricity  passing  through  the  galvanometer,  and  there- 
fore the  galvanometer  deflection,  is  proportional  to  this  E.M.F.  If 


FIG.  99 


the  observed  deflections  are  plotted  as  ordinates  and  the  corre- 
sponding values  of  the  angular  displacements  as  abscissas,  the 
smooth  curve  drawn  through  these  points  will  in  general  have 
practically  a  sine  form. 

89.  The  molecular  theory  of  magnetism.  The  phenomenon  of 
an  induced  E.M.F.  as  a  result  of  the  cutting  of  magnetic  lines  may 
be  made  use  of  to  determine  the  magnetic  condition  of  iron,  for 
example  the  distribution  of  magnetism  in  a  bar  magnet.  Before 
considering  the  method  of  doing  this,  it  is  desirable  to  consider 
the  general  nature  of  magnetism. 

It  is  a  matter  of  experimental  knowledge   that  into  however 
small  parts  a  magnet  is  divided  each  part  still  retains  magnetic 


ELECTRO-MAGNETIC   INDUCTION 


135 


properties.  The  assumption  which  naturally  follows  is  that  the  mol- 
ecules of  magnetic  metals  are  themselves  magnets.  This  hypothesis 
offers  an  easy  explanation  of  the  phenomenon  of  induced  mag- 
netism, —  that  is,  the  phenomenon  of  the  acquisition  of  mag- 
netic properties  hy  a  bar 
of  iron,  or  of  steel,  as  a 
result  of  its  presence  in 
a  magnetic  field.  For  we 
have  only  to  assume  that 
in  an  unmagnetized  bar  the  molecular  magnets  have  various  orien- 
tations, so  that  their  magnetic  effects  neutralize  one  another,  and 
consequently  the  bar  as  a  whole  shows  no  magnetic  properties. 
It  is  not  improbable  that  the  molecular  magnets  are  arranged 
in  small  groups,  as  shown  in  Figure  100,  the  magnets  of  each 
group  being  in  equilibrium.  When  the  bar  is  under  the  influence 
of  a  magnetic  field  this  condition  of  equilibrium  is  disturbed,  and 


FIG.  100 


FIG.  101 


equilibrium  does  not  again  exist  until  some,  at  least,  of  the  mole- 
cules have  formed  new  groupings,  oriented  in  the  general  direction 
of  the  impressed  magnetic  field.  If  the  field  is  very  intense,  the 
alignment  might  perhaps  be  well-nigh  perfect,  as  in  the  ideal 
diagram  of  Figure  101,  which  corresponds  to  a  condition  of  entire 
saturation;  but  in  an  ordinary  magnet  the  alignment  would  be  very 
imperfect,  and  doubtless,  also,  some  of  the  closed  molecular  groups 


136 


ELECTKICITY,  SOUND,  AND  LIGHT 


would  still  exist.  Figure  102  is  an  attempt  to  picture  something 
more  or  less  similar  to  this  condition.  When  the  directive  force 
of  the  field  is  withdrawn,  some  of  the  molecules  probably  form 
new  closed  groups,  but  a  large  number  retain  the  direction  into 
which  they  have  been  rotated  by  the  field.  The  fact  that  causes 
which  facilitate  a  rearrangement  of  the  molecules,  such  as  heating 
or  jarring  by  blows,  facilitate  also  either  the  induction  of  magnet- 
ism in  a  bar  placed  in  a  magnetic  field,  or  its  demagnetization  when 
withdrawn  from  the  field,  may  be  taken  as  evidence  in  favor  of 
this  hypothesis  as  to  the  molecular  nature  of  magnetism. 

90.  The  distribution  of  magnetism.  Since  through  the  space 
surrounding  a  magnet  the  magnetic  lines  of  force  are  known  to 
pass  from  its  N  end  to  its  S  end,  it  is  in  accordance  with  the 


I  ^N^5/Tj)) 

v  ^^^^^ 


FIG.  102 

molecular  hypothesis  to  assume  that  these  lines  form  closed  curves 
passing  through  the  iron  from  the  N  end  of  each  molecule  to  the  S 
end  of  the  next  molecule  (see  Fig.  101).  Wherever,  then,  lines  of 
force  leave  the  large  magnet  there  are  free  molecular  N  poles,  i.e. 
N  poles  of  molecules  not  in  immediate  conjunction  with  neighbor- 
ing S  poles  (see  Fig.  102).  And  similarly,  wherever  magnetic  lines 
enter  the  large  magnet  there  are  free  S  poles.  In  the  case  of  a 
bar  magnet  these  lines  of  force  leave  the  magnet  along  the  entire 
N  half  and  enter  it  along  the  entire  S  half  after  the  manner  shown 
in  Figure  102.  The  distribution  of  these  lines  may  then  be  taken 


ELECTRO-MAGNETIC  INDUCTION 


137 


as  a  measure  of  the  distribution  of  magnetism  along  the  bar. 
Within  the  bar  the  actual  number  of  lines  is  of  course  greatest  at 
the  center  where  the  largest  number  of  molecules  are  aligned  in 
the  direction  of  the  axis  of  the  magnet.  But  the  magnitude  of 
any  action  at  a  point  in  the  immediate  neighborhood  of  the  bar 
depends  upon  the  number  of  magnetic  lines  entering  or  leaving 
the  bar  at  that  point,  i.e.  upon  the  distribution  of  magnetism.  The 
distribution  is  not  uniform  along  either  half  of  the  bar,  and  except 
for  a  bar  which  is  carefully  magnetized  it  is  also  asymmetrical 
with  respect  to  the  center  of  the  bar. 

91.  Experimental  determination  of  the  distribution  of  mag- 
netism. The  distribution  of  magnetism  in  a  bar  magnet  may  be 
found  by  the  use  of  a  "  test  coil."  This  is  a  small  coil  c  (Fig.  103) 


FIG.  103 

which  fits  closely  over  the  bar  and  which  may  be  slipped  along  it, 
thus  cutting  the  lines  which  enter  or  leave  the  magnet.  The  ter- 
minals of  the  coil  are  connected  to  a  moving-coil  galvanometer, 
the  deflections  of  which  are  proportional  to  the  quantity  of  induced 
electricity  which  flows  through  the  galvanometer,  and  hence  to 
the  number  of  lines  entering  or  leaving  the  magnet  in  the  space 
over  which  the  coil  has  moved.  These  deflections  are  plotted  as 
ordinates  with  a  direction  up  or  down  depending  on  the  direction 
of  the  galvanometer  throw  (see  Fig.  104),  the  abscissas  corre- 
sponding to  these  ordinates  being  the  distances  from  one  end  of 
the  magnet  to  the  middle  of  the  space  over  which  the  test  coil  is 


138 


ELECTRICITY,  SOUND,  AND  LIGHT 


moved.    A  smooth  curve  is  then  drawn  through  these  points.    The 
intersection  of  this  curve  with  the  axis  of  distances  gives  the  dis- 
tance from  one  end  of  the  bar  to  the  point  of  no  free  magnetism, 
—  that  is,  to  the  magnetic  center  of  the  bar. 

92.  Location  of  the  poles  in  a  bar  magnet.  Now  the  moment 
of  force  which  would  act  upon  the  magnet  if  placed  at  right  angles 
to  a  uniform  magnetic  field  is  by  definition  proportional  to  the 
magnetic  moment  of  the  magnet  (p.  18).  It  is  the  sum  of  the 


N.Pole: 3=    abs.x  ord.  =  2120.  , 

ord.  =  270         .-.    2=7.9 


S.Pole:*    abs.x  ord.  =  2310.   . 
*    ord.  =  283 


length  of  magnet  =  23  cm. 
distance  between  poles  =  1C.2  cm. 
"  "  "      =  .71ensrth 


Test  coil   Deflection    Coil 
OX       +  46.  mm.     12.5 
0.5-1.5cm.+  37.  •"         13.5 
1.5-2.5 "      34.  "         14.5 
15.5 

27.  16.5 

24. 
22. 
18.3 
15.5 
10. 
5.6 
1. 


FIG.  104 

moments  due  to  all  the  separate  quantities  of  magnetism  in  the  bar ; 
that  is,  it  is  proportional  to  the  sum  of  the  products  of  all  these 
quantities  of  magnetism  by  their  lever  arms  taken  with  respect 
to  an  axis  at  the  magnetic  center  of  the  bar.  Now  the  ordinates  of 
the  curve  are  proportional  to  the  quantities  of  magnetism.  There- 
fore a  number  proportional  to  the  magnetic  moment  may  be  found 
by  adding  all  the  products  of  these  ordinates  by  their  correspond- 
ing lever  arms  as  found  from  the  curve.  The  distance  from  the 


ELECTRO-MAGNETIC   INDUCTION  139 

point  of  intersection  of  the  curve  with  the  axis  of  abscissas  to 
the  foot  of  any  orclinate  is  the  lever  arm  corresponding  to  the 
quantity  represented  by  that  ordinate.  But  by  definition  the  mag- 
netic moment  is  the  product  of  the  pole  strength  of  the  magnet 
and  the  distance  between  the  poles.  Now  the  sum  of  all  the 
ordinates  on  one  side  of  the  magnetic  center  is  proportional  to  the 
pole  strength.  Therefore  the  distance  between  the  poles  may  be 
found  by  dividing  the  number  found  as  above  to  be  proportional 
to  the  magnetic  moment  by  the  number  proportional  to  the  pole 
strength.  This  distance  between  the  poles  of  a  magnet  may  be 
any  fraction  whatever  of  the  length  of  the  bar.  In  a  permanent 
magnet  it  never  exceeds  five  sixths  of  the  length  and  is  usually 
much  less  than  this. 

EXPERIMENT  12 

(A)  Object.  To  plot  the  curve  showing  the  variation  of  the  E.M.F.  induced 
in  the  coil  of  an  ideal  dynamo  with  the  angular  displacement  of  the  coil. 

Directions.  By  connecting  to  the  galvanometer  terminals  a  battery  which 
is  short  circuited  through  a  piece  of  copper  wire,  find  the  direction  of  the 
current  which  corresponds  to  a  given  direction  of  deflection.  Connect  the 
galvanometer  to  the  dynamo  and  allow  the  coil  of  the  latter  to  turn  through 
a  ten-degree  interval.  Applying  to  this  motion  the  dynamo  rule,  note  the 
verification  of  the  rule.  Then  following  the  method  outlined  in  section  88, 
note  the  deflections  of  the  galvanometer  for  ten-degree  intervals  during 
one  complete  revolution  of  the  dynamo  coil.  Choose  as  the  axis  of  abscissas 
a  line  about  the  middle  of  the  sheet  of  coordinate  paper  and  in  the  direc- 
tion of  its  length.  Choose  the  scale  for  plotting  the  curve  as  large  as  the 
size  of  the  sheet  will  allow.  Tabulate  the  observed  throws  and  the  corre- 
sponding angular  displacements  in  one  corner  of  the  sheet  and  let  this  sheet 
be  the  record  of  the  experiment  (see  Fig.  105). 

(B)  Object.    To  plot  the  curve  representing  the  distribution  of  the  mag- 
netic lines  leaving  a  bar  magnet  and  to  find  the  actual  distance  between 
the  poles  of  the  magnet. 

Directions.  The  magnet  mm'  to  be  used  is  contained  in  a  slotted  brass 
tube  on  which  a  scale  is  ruled  or  pasted  (see  Fig.  103).  The  test  coil  c 
may  be  moved  suddenly  between  the  stops  e  and  f  of  the  frame.  These 
should  be  so  set  as  to  admit  of  a  motion  of  the  coil  of  about  1  cm.  For 
convenience  of  manipulation  this  distance  may  be  made  an  even  sub- 
t  multiple  of  the  length  of  the  magnet  (e.g.  T^). 

For  the  first  deflection  set  the  frame  so  that  e  and  f  are  equidistant 
from  the  right  end  of  the  magnet.  Now  remove  the  stop  /  and  the  test 


140 


ELECTBICITY,  SOUND,  AND  LIGHT 


coil  from  the  tube.  Then  note  the  deflection  caused  by  suddenly  slipping 
the  test  coil  back  over  the  end  of  the  magnet  to  a  position  against  the 
stop  e.  This  motion  cuts  all  the  lines  that  leave  the  end  of  the  magnet, 
and  the  corresponding  deflection  is  therefore  a  measure  of  the  magnetism 
at  the  end  of  the  bar.  Plot  it  from  the  point  on  the  X  axis  of  your  diagram 
which  corresponds  to  the  end  of  the  bar.  Replace  the  stop  f  and  shift  the 
frame  until  when  the  coil  is  against  /  it  is  in  exactly  the  same  position  as 
when  it  was  against  e  before.  Place  e  so  that  the  coil  moves,  say,  1  cm. 
between  stops,  then  move  it  quickly  from /to  e  and  plot  the  throw  at  the 
point  on  the  X  axis  which  corresponds  to  the  mid-point  of  its  motion. 


.  3  J  

:g  it  £-.-.i|: 

armature  : 

:::::::::::::     o*-10'  10mm.   140'  7mm.  280'    18.5  ::: 
"  10'-20°  14  «       150'  3.5       290"    16.     :: 

::Jj:::::::::::: 

:5::::::::::::::::|::^i 

41|lilii| 

;!Si!llffi!ij!!il!;i 

60'  22.          19010.5       330'      4.     - 
90'  21.          220'  19.         360*      7.0  -- 

IN                     120"  14.          250'  22. 
±:::                     130'  10.         260  '21.5 

^  :  2f  £  :  ~-  <  £  :  :  -  ^  :  :  H  £  2  Of  ;!!!$£  '2-  0!  :  |j  j&  :  :  iffi  :  :  )   !  :  : 

ii.  

^|::::::^±:X^|:::|::::::||::|^:: 

FIG.  105 

Following  the  method  outlined  in  section  91,  continue  until  the  opposite 
end  of  the  magnet  is  reached.  Then  remove  the  stop  e  and  take  the  deflec- 
tion caused  by  slipping  the  test  coil  entirely  off  the  tube. 

Plot  the  curve  showing  the  distribution  of  magnetism,  utilizing  in  so 
doing  the  full  size  of  the  sheet  of  coordinate  paper.  Tabulate  upon  the  same 
sheet  the  values  used  in  plotting  the  curve.  Also  indicate  on  this  sheet  the 
sum  of  the  products  of  the  ordinates  by  their  corresponding  abscissas, 
the  sum  of  the  ordinates,  and  the  value  determined  for  the  distance  be- 
tween the  poles.  Let  the  curve  and  these  tabulated  values  be  the  record 
for  the  experiment  (see  Fig.  104). 


CHAPTEE  XIII 


THE  CONSTANTS  OF  THE  EARTH'S  MAGNETIC  FIELD 

93.  Direction  of  the  earth's  magnetic  field.  A  magnetic  needle 
placed  in  the  earth's  magnetic  field,  and  free  to  rotate  only  in  a 
horizontal  plane,  i.e.  about  a  vertical  axis,  is  not  found,  in  general, 
to  set  itself  exactly  in  coincidence  with  the  geographical  meridian. 
The  angle  by  which  the  direction  of  the  needle  differs  from  the 
geographical  north-and-south  line  is  known  as  the  variation  at 
the  point  at  which  the  needle  is  placed. 

Again,  if  the  needle  is  free  to  rotate  about  a  horizontal  axis,  it 
is  not  found  to  assume  a  horizontal  position,  but  in  the  northern 
latitudes  the  N  end  dips  downward.  The  angle  which  H 

the  needle  makes  with  the  horizontal  at  any  point  is 
known  as  the  magnetic  dip  at  that  point.  For  most 
localities  within  the  United  States  the  angle  of  dip 
is  between  60°  and  75°. 

Let  the  intensity  of  the  earth's  field  be  denoted 
by  /.  Since,  in  most  of  the  instruments  in  which  a 
magnetic  needle  is  employed,  such,  for  example,  as 
the  tangent  galvanometer,  the  needle  is  suspended  so 
as  to  be  free  to  turn  only  in  a  horizontal  plane,  it 
is  usually  only  the  component  of  /  parallel  to  the 
horizon  which  it  is  necessary  to  determine.  In  the  experiment 
which  follows,  however,  we  shall  measure  both  the  horizontal 
component  H  and  the  vertical  component  V  of  the  earth's  field, 
and  from  these  measurements  we  shall  deduce  the  absolute 
intensity  I  of  this  field  in  the  direction  in  which  it  acts,  and 
the  angle  §  which  the  direction  of  this  field  makes  with  the 
horizontal.  The  relations  which  exist  between  these  four  quan- 
tities, and  by  means  of  which  these  deductions  can  be  made, 
are  obviously  the  following  (see  Fig.  106)- 

141 


FIG.  106 


142  ELECTRICITY,  SOUND,  AND  LIGHT 

V  —  I  sin  8. 


(1) 

(2) 

(3) 


The  quantities  IT,  V,  and  B,  with  the  variation,  are  known  as  the 
magnetic  constants  of  any  given  locality. 

94.  Absolute  measurement  of  H  and  V  by  means  of  an  earth 
inductor.  The  most  rapid  and  convenient  and,  on  the  whole,  one 
of  the  most  satisfactory  methods  of  measuring  H  or  V  is  based 
upon  the  principle  of  electro-magnetic  in- 
duction. It  requires  a  ballistic  galvanom- 
eter, a  resistance  box,  and  a 
"test  coil"  known  as  an  "earth 
inductor"  (see  Fig.  107).  Such 
an  instrument  consists  simply 
of  a  circular  coil  of  large  area 
and  large  number  of  turns,  so 
mounted  that  it  may  be  rotated 
very  suddenly  through  180°. 
The  principle  involved  is  that 
stated  in  section  85  (p.  130), 
namely,  that  the  E.M.F.  in- 
duced in  a  con- 
ductor is  equal  to 
the  rate  at  which 
the  conductor  cuts 
lines  of  magnetic 
force.  Thus  consider  the  coil  of  the  earth  inductor  to  have  an 
average  area  of  a  square  centimeters  and  to  be  composed  of  n 
loops  of  wire.  Let  it  be  placed  so  that  at"  the  beginning  of  the 
motion  its  plane  is  at  right  angles  to  the  horizontal  component 
of  the  earth's  field.  In  Figure  108  the  heavy  line  represents  the 
coil  in  this  position.  As  then  it  rotates  through  one  fourth  revo- 
lution about  a  vertical  axis  lying  in  the  plane  of  the  coil,  each 
loop  cuts  Ha  lines.  Its  position  is  then  represented  by  the  dotted 
lines  of  the  figure.  In  the  next  one  fourth  revolution  each  loop 


FIG.  107 


CONSTANTS  OF  EARTH'S  MAGNETIC  FIELD     143 


also  cuts  Ha  lines,  but  in  such  a  manner  that  the  E.M.F.  induced 
is  in  the  same  direction  as  that  induced  by  the  first  one  fourth 
revolution.  In  one  half  revolution,  then,  each  loop  cuts  2  Ha  lines. 
If  T  represents  the  time  of  this  half  revolution  of  the  coil, 
the  average  E.M.F.  induced  in  each  loop  is 


- — -•    Hence  the  average  E.M.F.  for  the 
coil  of  n  loops  is  -       —  •    The  product  an 


will  be  denoted  by  A  and  known  as  the 
total  area  of  the  coil.  If  the  terminals  of 
the  coil  are  connected  to  a  ballistic  gal- 
vanometer circuit  such  that  the  total  re- 
sistance of  the  circuit  is  R  C.G.S.  units, 
this  E.M.F.  causes  to  flow  a  current  of 

2  HA 

average  value  -  — —  •    The  quantity  Q  of  electricity  caused  to  pass 
RT 

through  the  galvanometer  is  T  times  this  current.    Hence 


i 
i 

FIG.  108 


1HA 
R 


C.G.S.  units. 


(4) 


If  the  time  r  is  so  small,  as  compared  with  the  period  of  the  gal- 
vanometer, that  this  quantity  Q  may  be  assumed  to  have  passed 
before  the  coil  turns  appreciably,  then  Q  may  be  obtained  from 
the  throw  BH  and  the  galvanometer  constants,  as  explained  in  sec- 
tion 61  (p.  93).  Thus  Q  =  —Lz£—  From  this  and  equation  (4) 


7T 


the  value  of  If  is  obtained  in  the  form 


RQ 


27TA 


(5) 


Both  p,  the  damping  factor,  and  t,  the  half  period  of  the  gal- 
vanometer, must  be  determined  under  the  conditions  prevailing 
in  the  circuit  at  the  time  of  the  observation  of  QH\  for  the  ratio 
p  [=  OJO^\  will  not  have  at  all  the  value  which  it  had  in  Experi- 
ment 8,  where  the  galvanometer  was  swinging  on  open  circuit. 
In  the  present  case  the  circuit  of  the  galvanometer  coil  is  at  all 
times  closed  through  the  resistance  R.  It  is  desirable  to  make  R 


144  ELECTEICITY,  SOUND,  AND  LIGHT 

so  large,  through  the  insertion  of  resistance  boxes  into  the  circuit, 
that  the  coil  will  make  at  least  ten  or  fifteen  swings  before  com- 
ing to  rest.  Otherwise  neither  p  nor  t  can  be  determined  with 
sufficient  accuracy. 

If  the  coil  is  turned  over  so  that  in  its  original  position  its 
plane  is  perpendicular  to  the  vertical  component  of  the  earth's 
field,  a  similar  set  of  observations  will  of  course  give  the  value 
of  V.  Thus 


If  we  do  not  care  to  determine  either  H  or  V  absolutely,  but  wish 
to  know  simply  the  value  of  the  angle  of  dip  S,  we  have  only  to 
take  the  throws  6H  and  Qv  corresponding  to  the  motions  of  the 
coil  which  cause  it  to  cut  the  horizontal  and  vertical  components 
respectively.  We  obtain,  then,  from  equations  (5)  and  (6), 

V  _QV 

r\ 

and  from  equation  (3),         S  =  tsm~1~-  (8) 

"H 

EXPERIMENT  13 

(A)  Object.    To  find  the  angle  of  dip. 

Directions.  Connect  in  series  the  ballistic  galvanometer,  a  resistance  box, 
and  the  earth  inductor,  the  last  instrument  being  placed  in  the  position 
occupied  by  the  tangent  galvanometer  in  Experiment  3.  Adjust  the  resist- 
ances of  the  box  until  the  throw  produced  by  cutting  the  horizontal  com- 
ponent of  the  earth's  lines  is  from  three  to  six  centimeters.  Take  the 
mean  of  six  throws,  three  to  the  right  and  three  to  the  left,  as  the  correct 
value  of  0H.  Take  similarly  six  observations  upon  Ov.  Then  compute  8 
from  equation  (8)  above.  Change  the  value  of  the  resistance  in  the  box 
and  see  how  well  you  can  duplicate  the  ratio  6y/0n. 

(B)  Object.  To  determine  the  absolute  values  of  V  and  //. 
Directions.  Insert  as  much  as  10,000  ohms  into  the  circuit.    Set  the  coil 

so  as  to  cut  the  vertical  component  of  the  earth's  field  and  take  observa- 
tions upon  Oy,  t,  and  p. 

Measure  the  resistance  of  the  earth  inductor  and  the  galvanometer, 
either  by  the  bridge  method  or  by  determining  the  E.M.F.  of  a  Daniell  or 
a  storage  cell  and  observing  with  a  milliammeter  the  current  which  this 
cell  sends  through  the  inductor  and  the  galvanometer  in  series  (see  Fig.  63, 


CONSTANTS  OF  EABTH'S  MAGNETIC  FIELD      145 

p.  84).  Either  determine  K  or  take  its  value  from  the  results  of  Experi- 
ment 7.  Measure  the  mean  area  a  of  the  inductor  and  multiply  by  the 
number  of  turns  to  find  A.  Compute  V  from  equation  (6),  remembering 
that  R  in  C.G.S.  units  is  R  in  ohms  x  109,  and  that  K  in  C.G.S.  units  is  K 
in  amperes  x  10"1. 

Having  found  V,  compute  H  by  multiplying  V  by  the  value  of  6H/0V 
found  in  (A). 

From  V  and  8  compute  7. 

EXAMPLE 

(A)  The  earth  inductor  was  placed  in  the  position  occupied  by  the 
tangent  galvanometer  in  Experiment  3.    When  there  was  no  additional 
resistance  in  the  galvanometer  and  earth-inductor  circuit,  the  throws  cor- 
responding to  V  and  H  were  20.51  cm.  and  6.53  cm.    Hence  tan  8  —  3.139. 
When  there  was  4000  ohms  additional  resistance,  making  the  total  4919 
ohms,  the  throws  were  6.35  and   2.00  cm.     Hence  tan  8  =  3.175.     The 
mean  value  of  3.157  corresponds  to  an  angle  of  72°  28'  30",  the  angle 
of  dip. 

(B)  Making   R    now  equal   to    10,919    ohms,    and   placing    the    scale 
146.7  cm.  from  the  galvanometer  mirror,  an  average  throw  of  3.30  cm. 
was  observed  when  the  vertical  component  of  the  earth's  field  was  being 
cut.    Hence  Qv—  .01125  radians.    The  period  and  damping  factor  of  the 
galvanometer  found  under   these    conditions  were   t  —  3.20   seconds   and 
pi  =  1.103.    The  constant  K  was   found  to  be,   in   amperes  per  radian, 
.00001055.     The   average   area  of  the   earth  inductor  was  208.7  sq.  cm. 
and  there  were  600  turns.    Hence 

=  10919  x  109x  .00001055  x  1Q-1  x  3.20  x  .01125  x  1.103  _  5g23 
27rx600  x  208.7 

Also       H  =  -^-  =  '-^  =  .1845       and       7  =  -^=.6108. 
tan  8      3.157  sin  8 

The  value  of  H  as  found  by  the  earth  inductor  agreed,  therefore,  to 
within  .6  per  cent  with  the  value  .1856  determined  in  Experiment  2,  for 
the  same  locality,  by  using  a  magnetometer. 


CHAPTEE  XIV 
SELF-INDUCTION 

95.  Nature  of  self-induction.  In  section  83  (p.  126)  the  neces- 
sary and  sufficient  condition  for  the  electro-magnetic  induction  of  an 
E.M.F.  in  a  conductor  was  found  to  be  the  cutting  of  magnetic  lines 
of  force  by  the  conductor.  It  follows,  then,  that  in  general  any  con- 
ductor which  lies  in  a  magnetic  field  will  have  an  E.M.F.  induced  in 
it  by  any  change  in  the  intensity  of  this  field,  for  such  a  change  must 
cause  lines  of  force  either  to  appear  or  to  disappear,  and  in  so  doing 

they  will,  in  general,  sweep  across 
the  conductor.  Now  since  a  straight 
conductor  carrying  a  current  lies  in 
its  own  magnetic  field  (see  Fig.  109, 
wThich  represents  a  section  of  the 
field  about  a  conductor  carrying  a 
current  into  the  plane  of  the  paper), 
and  since  the  strength  of  this  field 
varies  as  the  current  varies,  it  is  to 
be  expected  that  an  induced  E.M.F. 
will  be  set  up  in  every  element  of 
this  conductor  by  any  change  in  the 

current  which  it  carries.  This  inference  is  completely  confirmed  by 
experiment.  We  may  picture  this  so-called  E.M.F  of  self-induction 
to  arise  as  follows :  when  the  field  shown  in  the  figure  collapses, 
i.e.  when  the  current  in  the  wire  dies  out,  the  circular  Lines  of  force 
may  be  thought  of  as  shrinking  to  points  at  the  center  of  the  wire, 
and  in  so  doing  sweeping  across  the  conductor  from  outside  to  inside. 
Similarly,  when  the  current  rises  we  may  imagine  these  lines  as  spring- 
ing from  the  center  outward,  and  in  so  doing  sweeping  through  the 
conductor  from  inside  to  outside.  In  each  case,  in  accordance  with 
Lenz's  law,  the  induced  E.M.F.  must  be  in  such  a  direction  as  to 
oppose  the  change  which  is  inducing  it,  i.e.  in  such  a  direction  as 

146 


SELF-INDUCTION  147 

to  oppose  the  growth  of  the  current  when  the  latter  is  rising  and  to 
retard  its  decay  when  it  is  falling.  We  may  reach  this  conclusion 
also  by  the  direct  application  of  the  dynamo  rule,  remembering  that 
the  direction  in  which  the  conductor  is  cutting  the  lines  is  opposite 
to  the  direction  in  which  the  lines  are  moving  across  the  conductor. 

It  will  be  seen  that  the  self-induction  of  an  electrical  current 
is  exactly  analogous  to  the  inertia  of  a  mechanical  system,  for,  just 
as  inertia  is  the  property  of  such  a  system  by  virtue  of  which  it 
resists  any  attempt  to  change  its  condition  of  rest  or  of  motion,  so 
self-induction  is  the  property  of  an  electrical  system  by  virtue  of 
which  it  resists  any  attempt  to  change  its  existing  state. 

96.  Self-induction  in  various  forms  of  circuit.-  Since  self- 
induction  is  due  to  the  cutting  of  the  circuit  by  its  own  mag- 
netic lines,  it  will  be  seen  that  we  have  it  in  our  power  to 
increase  or  decrease  it  almost  B\ 
at  will,  for  we  can  easily  give  Sw 


the  circuit  such  a  form  that    ^-^ 
its  own  magnetic  lines  will  sweep  across  it  many  or  few  times. 
Thus  if  we  wish  to  remove  self-induction  entirely  from  a  circuit, 
we  have  only  to  arrange  it,  as  in  Figure  110,  so  that  the  outgoing 
and  incoming  currents  from  the  generator  B  travel  over  practically 
the  same  path.    In  this  case  there  will  be  no  magnetic  field  what- 
ever about  the  conductor,  since  the  fields  due  to  the  outgoing  and 
1 1  incoming  currents  are  everywhere  equal  and  opposite. 

|LT    Otherwise  stated:  if  the  current  changes,  the  E.M.F.  in- 
duced in  each  element  of  the  outgoing  conductor  by  the 
cutting  of  its  own  lines  is  exactly  neutralized  by  the  oppo- 
site E.M.F.  due  to  the  cutting  of  the  oppositely  directed 
lines  of  the  adjacent  element  in  the  other  conductor. 
The  coils  used  in  resistance  boxes  are  always  made  nonin- 
ductive  in  this  way.   The  wire  is  first  doubled  on  itself  and  then  wound 
on  the  spool  as  a  double  strand,  in  the  manner  shown  in  Figure  111. 
On  the  other  hand,  if  we  wish  to  make  a  circuit  of  very  large 
self-induction  we  have  only  to  arrange  it  in  the  form  of  a  closely 
packed  coil  of  many  turns,  through  all  of  which  the  current  passes  in 
the  same  direction  (see  Fig.  112),  for  in  this  case  the  lines  of  force 
which  the  rise  of  the  current  in  each  loop  would  thrust  through  that 


148  ELECTRICITY,  SOUND,  AND  LIGHT 

loop  are  also  thrust  through  all  the  other  loops,  so  that  the  E.M.F. 

of  self-induction  which  opposes  the  rising  current  is  enormously 

increased.  If,  for  example,  the  coil 
has  100  turns,  the  lines  of  force  due 
to  any  one  turn  sweep  across  the  en- 
tire 100  turns.  Across  any  turn,  then, 
there  sweeps  100  times  the  number 
of  lines  which  it  itself  produces,  and 
therefore,  other  tilings  being  equal, 
there  is  induced  in  each  turn  100 
times  the  E.M.F.  which  would  be  in- 

FIG.  112  ,         ,    .  .      , 

duced  in  a  single  turn.    But  since 

there  are  100  turns  in  series,  the  E.M.F.  of  self-induction  in  the  whole 
coil  is  1002  times  as  large  as  it  would  be  for  a  coil  of  a  single  turn. 
97.  Units  of  self-induction.  It  is  evident  from  the  last  section 
that  the  total  number  of  cuttings  by  lines  of  force  which  a  given 
conductor  undergoes  because  of  the  appearance  or  disappearance 
of  a  given  current  within  it  depends  upon  the  form  of  the  con- 
ductor. For  a  given  conductor  it  is  also  directly  proportional  to 
the  current,  since  the  total  number  of  lines,  i.e.  the  magnetic  field 
strength,  about  the  conductor  is  proportional  to  the  current  within 
it.  The  factor  by  which  the  current  must  be  multiplied  to  give 
the  number  of  cuttings  is  then  a  constant  of  the  conductor.  It  is 
called  the  coefficient  of  self-induction  of  the  conductor.  If  L  repre- 
sents this  coefficient  for  a  given  conductor,  /  the  current  flowing 
through  it,  and  N  the  total  number  of  cuttings  by  lines  of  force 
which  the  conductor  experiences  when  this  current  is  stopped, 
then  the  definition  of  the  coefficient  of  self-induction  is  given  by 
the  equation 

N=LI         or         £  =  — •  (1) 

In  words,  the  coefficient  of  self-induction  of  a  conductor  may  be 
defined  as  the  total  number  of  times  that  this  conductor  is  cut  l>y 
lines  of  force  when  the  current  which  it  carries  undergoes  an 
increase  or  a  decrease  of  one  unit. 

From  equation  (1)  it  is  obvious  that  L  is  1  when  I  is  1  and  N 
is  1 ;  i.e.  the  absolute  unit  of  self-induction  is  the  self-induction  of 


SELF-INDUCTION 


149 


coefficient 


a  conductor  which  is  cut  once  by  a  line  of  force  when  the  current 
within  it  undergoes  a  change  of  one  absolute  electro-magnetic  unit 
of  current.  The  practical  unit  of  self-induction,  the  henry,  named 
in  honor  of  the  American  physicist  Joseph  Henry  (1799-1850),  is 
taken  as  109  times  this  unit.  It  is  the  self-induction  of  a  conductor 
which  is  cut  by  lines  of  force  109  times  when  the  current  within 
it  undergoes  a  change  of  one  absolute  unit,  or  by  108  lines  of  force 
when  the  current  undergoes  a  change  of  one  ampere  (see  also  sect. 
98).  If  this  cutting  takes  place  in  just  one  second,  the  mean  E.M.F. 
of  self-induction  developed  is  evidently  one  volt  (see  sect.  8 5,  p.  129). 
98.  The  measurement  of  self-induction.  One  of  the  simplest 
ways  of  determining  the  coefficient  of  self-induction,  of  a  coil 
of  wire  for  which 
this  coefficient  is 
not  too  small  is 
the  following.  The 
coil  8  (Fig.  113), 
the  coefficient  of 
which  is  sought, 
is  joined  in  series 
with  a  variable 
noninductive  re- 
sistance rx,  and  S 
and  rl  are  then 
made  one  arm  of 
a  Wheatstone's 

bridge,  the  other  arms  of  which,  M,  0,  and  T,  are  also  noninductive 
resistances.  After  suitable  values  have  been  given  to  M,  0,  and  T,  r1 
is  adjusted  until,  when  K^  is  first  closed  and  then  K2,  the  galva- 
nometer shows  no  deflection.  This  is  of  course  merely  the  balanc- 
ing of  resistances  for  steady  currents,  such  as  has  already  been  done 
in  connection  with  the  Wheatstone's  bridge.  If,  however,  while  7T2 
is  closed,  K^  is  suddenly  opened,  the  condition  of  steady  currents  no 
longer  holds.  Since  rl ,  T,  0,  and  M  are  noninductive  resistances, 
the  decaying  current  can  cause  no  cutting  of  lines  by  these  con- 
ductors. In  the  case  of  S,  however,  the  interruption  of  the  current 
causes  the  field  about  S  to  collapse,  and  there  is  thus  induced  in  S 


FIG.  113 


150 


ELECTRICITY,  SOUND,  AND   LIGHT 


an  E.M.F.  which  causes  a  momentary  current  to  flow  through  the 
circuit.  This  circuit  is  composed  of  the  series  resistances  S,  rv  M, 
and  the  divided  circuit,  one  branch  of  which  is  G  and  the  other 
T  +  0.  If  we  let  R  represent  the  total  resistance  of  this  divided 
circuit,  then  the  total  resistance  of  the  circuit  through  which  the 
self-induced  current  flows  is  S  +  i\  +  R,  +  M.  If  r  represents  the 
time  required  for  the  disappearance  of  these  magnetic  lines  in  S, 
then,  since  the  induced  E.M.F.  is  numerically  equal  to  the  rate  of 
cutting  of  these  lines  of  force,  we  have 


the  mean  E.M.F.  of  self-induction  =  —  =  -^-- 


the  mean  current  of  self-induction  = 


LI 


(2) 


If  then  Ql  represents  the  total  quantity  of  electricity  which  flows 
through  the  circuit  when  these  L I  lines  of  force  cut  it  because  of  the 
stopping  of  the  current  /,  we  have,  since  quantity  is  the  product  of 
current  by  time,  T  r 

/) _L  /q\ 

^1   Tf />      i     ~T\T  \     / 

The  fraction  of  this  quantity  which  passes  through  the  galva- 
nometer G  is  of  course  determined  solely  by  the  character  of 

the  conductors  T, 
0,  and  G. 

The  coil  S  is  now 
replaced  by  a  con- 
denser C  (Fig.  1 1 4) , 
across  which  is 
shunted  a  noniii- 
ductive  resistance 
U.  U  is  then  given 
such  a  value  that 
a  balance  is  again 
obtained  when  Kv 
is  first  closed,  then 

/C.   This  means,  of 
FIG.  114 

course,   that    U_  = 

S+rr  The  quantity  Q  of  electricity  which  is  now  upon  the 
condenser  plates  is  the  capacity  C  of  the  condenser  times  the 


SELF-INDUCTION  151 

RD.  to  which  it  is  charged.  But  this  P.D.  is  by  Ohm's  law  U  X  /. 
When  now  Kl  is  opened,  A"2  being  kept  closed,  this  quantity  UC1 
discharges  partly  through  U  and  partly  through  R  and  M.  By  the 
law  of  parallel  connections  (see  eq.  11.,  p.  83),  the  fraction  Q2  of 
Q  which  follows  the  latter  path  is  given  by 


R+M      U  +  lt 


(4) 


Since  the  fraction  of  Q2  which  passes  through  the  galvanometer  is 
the  same  as  the  fraction  of  Ql  which  passed  through  it  in  the  dis- 
charge of  the  coil,  and  since  these  fractions  are  proportional  to 
the  two  throws  0l  and  02  of  the  galvanometer,  we  have 

e,=  Ql  =  LI  U+X  +  M. 

02      Q,      S+r^  +  X  +  M  U*C1 

or,  since  S  +  ?\  has  been  made  equal  to  U,  we  have 

z  =  r'c|»-  (6) 

We  have  then  but  to  know  the  shunt  resistance  U,  the  capacity  C, 
and  the  two  throws  6l  and  02,  in  order  to  determine  the  coefficient 
of  self-induction  L. 

If  U  and  C  are  expressed  in  absolute  electro-magnetic  units, 
then  L  will  also  be  expressed  in  absolute  units  of  self-induction. 
If  U  and  C  are  expressed  in  practical  units,  i.e.  U  in  ohms  and  C 
in  farads,  then  L  will  be  expressed  in  practical  units  of  self-induc- 
tion, i.e.  in  henrys.  Since  an  ohm  is  109  and  a  farad  10~9  abso- 
lute units,  it  is  clear  from  equation  (6)  that  a  henry  is  109  absolute 
units  ;  i.e.  that  a  henry  is  the  self-induction  of  a  conductor  which 
is  cut  by  108  lines  of  force  when  there  is  a  change  of  1  ampere  in 
the  current  carried  by  it  (see  also  sect.  97). 

99.  Practical  illustrations  of  self-induction.  One  of  the  most 
striking  experimental  proofs  of  the  existence  of  an  E.M.F.  of  self- 
induction  is  the  following.  If  a  100-volt  lamp  L  is  shunted  as 
in  Figure  115  across  an  electro-magnet  U,  which  is  energized  by  a 
25-volt  or  even  a  50-volt  battery,  the  lamp  will  not  glow  appre- 
ciably so  long  as  the  key  K  is  closed,  but  when  K  is  opened  the 


152 


ELECTRICITY,  SOUND,  AND  LIGHT 


E.M.F.  of  self-induction  will  force  a  large  enough  momentary  cur- 
rent through  the  lamp  to  make  it  glow  brightly.  Furthermore, 
if  K  is  opened  very  suddenly,  as  by  a  blow  struck  with  a  mallet, 

the  brilliancy  of  the  momentary  glow 
will  be  very  greatly  increased.  The 
experiment  furnishes  very  satisfac- 
tory demonstration  of  the  fact  that 
an  induced  E.M.F.  is  proportional  to 
the  rate  of  cutting  of  lines  of  force. 
For  when  K  is  opened  slowly  the 
E.M.F.  of  self-induction  causes  a 
spark  to  jump  between  the  opening 
contact  points,  and  as  these  points 
draw  farther  and  farther  apart,  this 
spark  is  drawn  out  into  an  arc  of 
gradually  increasing  resistance.  Since, 
on  account  of  the  formation  of  this 
arc,  the  current  dies  out  slowly  rather 
than  suddenly  as  K  is  opened,  the 
induced  E.M.F.  is  small.  Increasing 
the  suddenness  of  the  break  decreases 

the  time  required  for  the  magnetic  field  to  collapse,  and  hence 
increases  the  rate  of  cutting  of  the  circuit  by  lines  of  force ;  i.e.  it 
increases  the  E.M.F.  of  self- 
induction. 

100.  The  induction  coil. 
One  of  the  finest  illustrations 
of  self-induction  is  found  in 
the  operation  of  an  induc- 
tipn  coil.  A  primary  coil  P 
(Fig.  116)  is  wound  upon  a 
soft  iron  core  and  connected 
into  the  circuit  of  a  battery  B 
through  the  contact  point  o. 

A  secondary  coil  S  is  wound  about  the  same  core  as  the  primary. 
A  condenser  C  is  connected  across  the  spark  gap  o.  When  the  cur- 
rent starts  in  the  primary  it  magnetizes  the  core  and  draws  the  iron 


FIG.  115 


,( 


FIG.  116 


SELF-INDUCTION  153 

hammer  Ji  away  from  the  contact  point  o.  This  breaks  the  circuit 
and  the  spring  sp  then  restores  contact  at  o.  The  operation  then 
begins  over  again.  At  every  break  of  the  primary  at  o  a  spark 
passes  between  the  terminals  of  the  secondary,  but  no  spark  passes 
at  make.  The  cause  of  this  unidirectional  character  of  the  dis- 
charge between  n  and  n  is  found  in  the  fact  that  at  make  the 
effect  of  the  self-induction  of  the  primary  is  simply  to  retard  the 
growth  of  the  primary  current,  and  hence  to  render  the  growth  of 
the  field  within  the  secondary  slow.  This  means,  of  course,  a 
small  induced  E.M.F.  in  the  secondary.  At  break,  however,  the 
presence  of  the  condenser  causes  the  current  to  fall  to  zero  in  an 
exceedingly  short  time.  For  as  soon  as  the  contact  points  begin  to 
separate,  the  self-induction  of  the  primary,  which  would  normally 
create  an  arc  between  the  points,  now  drives  its  induced  current 
into  the  condenser  for  an  instant,  and  thus  gives  the  contact  points 
time  to  get  so  far  apart  that,  by  the  time  the  condenser  is  charged, 
the  spark  can  no  longer  leap  across  the  gap  and  set  up  an  arc. 
The  function  of  the  condenser  is  then  to  prevent  sparking  at  o 
and  thus  to  make  a  sudden,  rather  than  a  slow,  collapse  of  the 
field.  Thus,  while  the  same  number  of  lines  of  force  cut  the 
secondary  at  make  as  at  break,  they  take,  at  maket  perhaps  a  thou- 
sand times  as  long  to  do  it,  and  hence  the  E.M.F.  at  make  is  much 
too  weak  to  force  a  spark  across  the  terminals  of  the  secondary, 
unless  these  terminals  are  extremely  close  together,  for  example 
.01  of  an  inch. 

101.  The  transformer.  The  transformer  is  a  modified  form  of 
the  induction  coil.  In  it  the  core  to  be  magnetized  by  the  primary 
current  is  given  some  shape  such  that  the  magnetic  lines  of  force 
have  a  continuous  iron  path  instead  of  being  obliged  to  push  out 
into  the  air,  as  in  the  induction  coil.  Further,  it  is  an  alternating 
rather  than  an  intermittent  current  which  is  sent  through  the 
primary  of  a  transformer.  The  effect  of  such  a  current  is  first  to 
magnetize  the  core  in  one  direction,  then  to  demagnetize  it,  then 
to  magnetize  it  again  in  the  opposite  direction,  and  so  on.  These 
changes,  of  course,  induce  in  the  secondary  an  alternating  current 
similar  to  that  which  is  being  sent  through  the  primary.  The 
E.M.F.  induced  in  the  secondary  is  obviously  proportional  to  the 


154 


ELECTRICITY,  SOUND,  AND  LIGHT 


number  of  turns  of  wire  upon  it,  for  each  turn  is  threaded  by  all 
of  the  lines  which  pass  through  the  core. 

In  practice  the  transformer  is  largely  used  as  a  "  step-down 
transformer " ;  that  is,  the  mean  potential  at  which  the  primary 
is  supplied  is  many  times  higher  than  the  mean  potential  which 
is  induced  in  the  secondary.  In  this  case  the  primary  coil  has  a 
large  number  of  turns  and  the  secondary  coil  has  a  smaller  num- 
ber. For  example,  if  the  main  conductors  are  kept  at  a  mean  P.D. 
of  1100  volts,  while  the  lamps  of  the  secondary  require  110  volts, 
there  must  be  10  times  as  many  turns  upon  the  primary  as  upon 
the  secondary. 

Transformers  are  usually  kept  connected  in  parallel  to  the  high 
potential  mains  which  connect  with  the  alternating-current  dynamo, 


n 

/                  i 

\ 
( 

Dynamo 

a 

feP 

^ 

\                      Transformers 

0 

*gii 

-o 

ft  0  0 


Low  Potential  Mains  with  Lamps 
FlG.  117 


-o 

-o 
-o 


and  lamps  in  the  secondary  circuit  are  turned  on  as  needed  (see 
Fig.  117).  It  may  seem  at  first  sight  as  if,  with  this  arrangement, 
large  currents  would  always  be  flowing  through  the  primaries, 
whether  any  were  required  in  the  secondaries  or  not.  This  is, 
however,  not  the  case,  for  when  all  the  lamps  on  a  given  secondary 
are  turned  out,  so  that  no  current  is  delivered  by  it,  the  self-induc- 
tion of  the  primary  "  chokes  off  "  practically  all  current  from  this 
primary  itself.  For,  since  the  effect  of  self-induction  in  any  circuit 
is  simply  to  retard  the  growth  of  a  rising  current,  or  the  decay  of 
a  dying  current,  when  an  alternating  E.M.F.  is  applied  to  a  circuit 
of  sufficiently  large  self-induction,  the  current  due  to  the  impressed 
E.M.F.  in  one  direction  scarcely  gets  started  before  the  reversal 
conies ;  and  this  reversed  E.M.F.  in  its  turn  only  begins  to  change 


SELF-INDUCTION  155 

this  current  when  another  change  in  direction  occurs.  So  that,  in 
general,  a  circuit  of  sufficiently  large  self-induction  behaves  toward 
an  alternating  E.M.F.  just  as  a  body  of  sufficiently  large  mass 
behaves  toward  a  rapidly  alternating  mechanical  force;  i.e.  in 
neither  case  is  any  appreciable  effect  produced  upon  the  system. 
But  suppose  that  lamps  are  turned  on  in  the  secondary  of  such 
a  transformer  circuit.  The  currents  which  are  induced  in  the 
secondary  are,  by  Lenz's  law,  in  such  direction  as  at  all  times 
to  thrust  lines  of  force  through  the  core  in  a  direction  opposite  to 
the  direction  in  which  the  primary  E.M.F.  is  attempting  to  thrust 
lines  through  the  core.  The  actual  number  of  lines  sent  through 
the  core  by  the  rise  of  a  given  current  in  the  primary  will  then 
be  less  if  the  secondary  is  closed  than  it  will  be  if  the  secondary 
is  open.  Closing  the  secondary  circuit  is  therefore  equivalent  to 
withdrawing  a  certain  amount  of  self-induction  from  the  primary. 
Hence  a  larger  current  flows  through  the  primary  when  a  lamp  is 
turned  on  in  the  secondary  than  when  the  latter  is  open.  Turning 
on  two  lamps  in  the  secondary  withdraws  twice  as  much  self- 
induction  from  the  primary  and  thus  increases  in  like  amount  the 
current  which  the  impressed  E.M.F.  sends  through  it,  and  so  on. 
Thus  the  current  taken  from  the  mains  by  the  primary  of  a  trans- 
former automatically  adjusts  itself  to  the  demands  of  the  second- 
ary. If  no  energy  is  taken  from  the  secondary,  the  primary  takes 
practically  no  current  from  the  dynamo ;  but  if  a  large  amount 
of  energy  is  demanded  by  the  secondary,  a  large  current  is  taken 
from  the  dynamo  by  the  primary.  The  transformer  is  therefore 
merely  a  device  for  transferring  energy  from  one  circuit  to  another 
without  the  intervention  of  any  mechanical  motions  of  any  sort. 
Its  efficiency  is  commonly  as  high  as  97  per  cent,  the  remaining 
3  per  cent  being  transformed  into  heat  in  the  core  and  coils  of  the 
transformer. 

EXPERIMENT  14 

Object.    To  find  the  coefficient  of  self-induction  of  a  coil. 

Directions.  In  order  to  avoid  the  necessity  of  obtaining  a  perfect  balance 
for  steady  currents,  —  a  tedious  operation  at  best,  —  modify  as  follows 
the  ideal  arrangement  described  above.  Insert  a  double  key  Ki  as  in  the 


156 


ELECTRICITY,  SOUND,  AND   LIGHT 


diagram  (Fig.  118),*  so  that  when  K1  is  depressed  and  the  switch  K2  closed, 
the  battery  circuit  is  closed  through  the  bridge,  and  at  the  same  time  the 
galvanometer  is  short-circuited ;  and  also  so  that  when  K2  is  closed,  lifting 
the  finger  from  7ix  opens  the  short  circuit  on  the  galvanometer  an  instant 
before  it  opens  the  battery  circuit.  Give  the  resistances  AT,  0,  and  T  val- 
ues which  are  of  about  the  same  order  of  magnitude  as  the  galvanometer 
resistance.  Observe  the  zero  of  the  galvanometer,  then  close  in  succession 
Kv  the  lower  contact  of  Kv  and  then  the  upper  contact  of  Kr  Adjust  r1?  all 
these  contacts  being  kept  closed,  until  the  galvanometer  stands  approxi- 
mately at  its  zero.  If  now  opening  Kz  produces  a  large  deflection,  adjust 


FIG.  118 


rl  still  further  until  this  deflection  is  reduced  to  not  more  than  a  centi- 
meter or  two.  In  order  to  effect  this  adjustment  it  is  usually  necessary  to 
make  the  wire  which  connects  rl  to  T  a  few  feet  of  bare  German  silver 
wire  (say  No.  22),  and  to  slip  this  along  through  the  binding  post  of  rl  in 
such  a  way  as  to  include  a  greater  or  smaller  length  of  it  in  the  branch 
which  contains  rr  When  the  balance  has  been  made  correct  to  within 
a  centimeter  close  K2  permanently,  take  the  reading,  then  open  Kl  and 
read  the  throw.  Repeat  several  times  ;  then,  in  order  to  eliminate  thermal 
effects,  reverse  the  battery  terminals  with  a  commutator  or  otherwise, 
and  read  the  throw  in  the  opposite  direction.  Take  the  means  of  several 

*  This  scheme  of  connections  is  due  to  Fleming. 


SELF-INDUCTION  157 

throws  in  opposite  directions  as  6r  This  throw  will  be  practically  all  due 
to  the  self -inductive  current  from  S,  since  the  interval  between  the  open- 
ing of  the  galvanometer  short  circuit  in  Kl  and  the  battery  circuit  is  so 
brief  that  the  galvanometer  coil  has  no  appreciable  impulse  given  to  it 
because  of  the  want  of  exact  balance  in  the  bridge  arms.  This  statement 
may  be  tested  by  seeing  whether  the  same  throw  is  not  obtained  when  the 
balance  is  exact  as  when  it  is  inexact.  If  01  is  not  a  throw  of  several 
centimeters  at  least,  the  number  of  dry  cells  in  B  should  be  increased. 

Now  replace  S  and  rl  by  C  and  U.  Make  U  the  same  as  M,  0,  and  T, 
and  adjust  still  further  with  the  German  silver  wire,  if  necessary,  until  as 
good  a  balance  is  obtained  as  before. 

Take  the  throw  02  precisely  as  0l  was  taken,  and  substitute  in  equa- 
tion (6)  to  obtain  the  coefficient  of  self-induction  of  S.  This  will  be 
expressed  in  henrys  if  C  is  in  farads  and  U  in  ohms.  It  is,  of  course, 
desirable  to  make  Ol  and  02  of  the  same  order  of  magnitude.  If  02  is  too 
small,  use  a  condenser  of  larger  capacity.  If  it  is  too  large,  place  C 
in  shunt  with  a  fraction  only  of  U  and  use  the  number  of  ohms  across 
which  C  is  shunted  as  the  resistance  to  substitute  in  equation  (6). 


EXAMPLE 

The  coil,  the  self-induction  of  which  was  determined,  was  3.7cm.  high, 
9.9  cm.  in  external  diameter,  4.1  cm.  in  internal  diameter,  and  was  wound 
with  about  4000  turns  of  No.  25  copper  wire.  The  resistance  of  the  coil  was 
about  29  ohms,  that  of  the  galvanometer  about  600  ohms.  The  resistance 
in  each  of  the  four  branches  of  the  bridge  was  made  1000  ohms.  The  mean 
value  of  01  was  27.9  mm.,  the  current  being  furnished  by  three  dry  cells. 
The  condenser  used  was  of  -'-microfarad  capacity  and  produced  a  deflec- 
tion, when  U  was  1000  ohms,  of  45.3  mm.  Therefore  L  =  .205  henrys. 
This  differed  by  less  than  half  of  one  per  cent  from  the  value  marked 
upon  the  coil,  which  had  been  previously.obtained  by  a  careful  comparison 
with  a  standard  of  self-induction  by  a  method  similar  to  that  described  in 
section  67  for  comparing  condensers. 


CHAPTER  XV 


MAGNETIC  INDUCTION  IN  IRON 

102.  The  test  coil.  In  section  94  a  method  was  described  for 
measuring  the  intensity  of  a  magnetic  field  by  the  use  of  a  ballis- 
tic galvanometer  and  a  test  coil,  known  in  that  experiment,  from 
its  special  application  to  the  measurement  of  the  earth's  magnetic 
field,  as  the  "  earth  inductor."  The  principles  involved  may  be 
applied  to  a  general  study  of  magnetic  induction  and  are  here  re- 
stated with  particular  reference  to  the  experiment  which  follows 

this  section. 

Consider  a  magnetic 
field  which  may  be  es- 
tablished or  destroyed  at 
will  by  making  or  break- 
ing the  circuit  of  a  con- 
ductor C.  In  the  neigh- 
borhood of  this  conductor 
(see  Fig.  119)  there  is 
placed  a  small  coil  of 
wire  c,  the  test  coil.  Let  the  average  area  of  this  coil  be  such 
that  when  multiplied  by  the  number  of  turns  of  the  coil  the 
total  area  is  A  square  centimeters.  The  coil  is  connected  to  a 
ballistic  galvanometer,  the  constants  of  which  are  known.  The 
combined  series  resistance  of  the  galvanometer  and  coil,  in  C.G.S. 
units,  will  be  represented  by  R.  If  in  the  time  r  the  number 
of  magnetic  lines  passing  through  the  test  coil  is  changed  from 
zero  to  &C  lines  per  square  centimeter  by  the  establishment  of 
the  magnetic  field,  then  the  mean  E.M.F.  induced  in  this  coil 
during  the  time  r  is  cKA/T  C.G.S  units.  The  quantity  of  elec- 
tricity Q  which  this  E.M.F.  causes  to  flow  through  the "  galva- 
nometer is  given  by  the  relation  Q  =  &CA/R.  Q  can  be  found 

158 


FIG.  119 


MAGNETIC   INDUCTION  IN  IKON  159 

as  shown   in   section  94   from   the   throw   of   the   galvanometer. 
Transforming  the  expression  gives 


that  is,  QR/A  is  a  measure  of  the  number  of  magnetic  lines  per 
square  centimeter  passing  through  the  test  coiL 

103.  Magnetic  field-intensity.    Now  the  number  of  lines  passing 
through  the  test  coil  as  determined  from  the  galvanometer  throw 
is   found  to  depend  upon  the   medium  within  the   coil.     If  the 
medium  is  air,  the  number  of  magnetic  lines  per  square  centimeter 
is  known  as  the  magnetic  field-intensity.    And  even  if  the  medium 
is  not  air,  the  number  which  would  exist  were  the  medium  replaced 
by  air  is  still  known  as  the  magnetic  field-intensity  and  is  com- 
monly denoted  by  the  symbol  di  . 

104.  Diamagnetic  and  paramagnetic  substances.    The  number 
of  magnetic  lines  per  square  centimeter  determined  as  above  is  for 
most  substances  very  slightly  less  than  it  is  for  air.    Substances 
for  which  this  is  true  are  called  diamagnetic.    That  the  difference 
is  very  slight  may  be  seen  from  the  fact  that  for  bismuth,  the 
most  highly  diamagnetic  substance  known,  the  ratio  of  the  num- 
ber of  lines  established  to  the  number  which  would  be  established 
in  air  by  the  same  cause  is  approximately  .9998. 

Those  substances  for  which  the  number  of  lines,  other  things 
being  equal,  is  greater  than  it  is  in  air,  are  called  paramagnetic. 
Most  of  these  substances  also  differ  but  very  slightly  from  air. 
The  exceptions,  iron,  nickel,  and  cobalt,  with  certain  of  their  com- 
pounds or  alloys,  admit  of  such  exceptionally  large  values  for  the 
number  of  magnetic  lines  that  they  really  constitute  a  separate 
group.  In  comparison  with  members  of  this  group,  which  will 
be  known  as  magnetic,  all  other  substances  may  be  considered  as 
magnetically  indifferent,  and  for  practical  purposes  identical  in 
their  behavior  with  air. 

105.  Magnetic  induction.    The  name  magnetic  induction  is  given 
to  this  phenomenon  of  the  establishment  in  a  magnetic  substance 
of  a  number  of  magnetic  lines  different  from  the  number  which 
would  exist  in  air  from  the  same  cause.    To  study  this  phenomenon 


160 


ELECTKICITY,  SOUND,  AND  LIGHT 


consider  two  identical  coils  of  wire,  the  first  wound  about  an  annu- 
lar ring  composed  of  some  magnetic  material  (e.g.  soft  iron),  and 
the  second  about  an  identical  ring  of  some  magnetically  indifferent 
material  (e.g.  wood)  (see  Fig.  120).  These  two  coils  are  connected 
in  series  so  that  identical  conditions  of  current,  and  consequently 
of  magnetic  field,  exist  for  both  coils.  The  magnetic  conditions 
within  these  rings  are  observed  by  similar  test  coils  S  and  Sr. 
Upon  completing  the  circuit  through  the  primaries  the  number  of 
magnetic  lines  within  the  wood  core,  as  measured  by  the  expres- 
sion QR/A,  gives  the  magnetic  field-intensity  &C  in  lines  per  square 
centimeter.  The  test  coil  wound  over  the  iron  core  has  induced  in 


FIG.  120 

it  a  quantity  of  electricity  Q',  which  is  larger  than  Q,  and  which, 
when  substituted  in  the  expression  Q'R/A,  gives  a  measure  of 
the  number  of  lines  per  square  centimeter  through  the  iron.  This 
number  will  be  represented  by  the  symbol  cB  and  will  be  known 
as  the  induction  in  the  iron  due  to  the  magnetic  field  of  intensity  c?f. 
That  is,  induction  is  the  total  number  of  lines  per  square  centimeter 
passing  through  a  substance  as  a  result  of  its  presence  in  a  mag- 
netic field.  Induction  is  also  frequently  known  as  flux  density, 
since  it  follows  from  its  definition  that  it  is  the  flux  (or  total  num- 
ber of  lines  through  the  iron)  divided  by  the  area  of  cross  section 
normal  to  their  direction. 


MAGNETIC  INDUCTION  IN  IRON 


161 


106.  Magnetization.  If  a  larger  current  is  used,  a  larger  value 
of  cK  is  obtained,  and  a  larger  value  of  cS.  In  this  way  it  is  possi- 
ble to  study  the  variation  of  the  induction  in  iron  with  the  mag- 
netic intensity  of  field  which  causes  it.  The  curve  representing 
the  general  nature  of  the  variation  is  shown  in  Figure  121.  Now, 


lines  per  square  centimeter  in  the   air  core, 


since  there  are  <. 
there  must  be  3V 
of  the  lines  in  the 
iron  core  which 
are  due  to  the 
current.  The  dif- 
ference, cB  —  cTf , 
represents,  then, 
the  number  of 
lines  per  square 
centimeter  which 
are  due  solely  to 
the  presence  of 
the  iron  in  the 
magnetic  field  in- 
side the  circular 
coil.  It  is  taken 
as  a  measure  of 
the  magnetization 
of  the  iron.  The 

technical  definition  of  magnetization,  commonly  denoted  by  o7,  is 
this  number  of  lines  divided  by  4  TT.    Thus,  by  definition, 

/  fT>  «7/*\ 

(2) 


=  Field-Intensity 
FIG.  121 


4-7T 

The  reason  for  the  introduction  of  the  factor  4  TT  will  be  discussed 
in  a  later  paragraph. 

107.  Three  stages  of  magnetization.  When  successive  values 
of  c7  are  computed  from  the  last  equation  and  successive  values  of 
cB  and  cfC  as  given  in  Figure  121,  and  when  then  the  successive 
values  of  3  are  plotted  as  ordinates  with  the  corresponding  values 
of  c?f  as  abscissas,  a  curve  of  the  type  shown  in  Figure  122  is 


162 


ELECTRICITY,  SOUND,  AND  LIGHT 


obtained.  This  latter  curve  shows  that  the  process  of  magnetiz- 
ing iron  may  be  divided -in  to  three  stages  which  are  represented 
respectively  by  the  portions  of  the  curve  from  &  to  c,  from  c  to  d, 
and  from  d  to  e.  During  the  first  stage,  which  corresponds  to  very 
small  values  of  the  magnetizing  force  e?f,  the  magnetization  increases 
in  nearly  direct  proportion  to  c?f ;  then  during  the  second  stage, 
namely  from  c  to  d,  the  magnetization  of  the  iron  increases  exceed- 
ingly rapidly,  and  during  the  final  stage,  from  d  to  ey  it  becomes 
practically  constant,  showing  that  the  magnetization  of  iron  can- 
not be  pushed  higher  than  a  certain  limit,  however  strong  fields 

may    be    used. 
e  The   explanation 

of  this  behavior 
in  terms  of  the 
molecular  theory 
(see  sect.  89,p.  134) 
is  that  while  the 
field  is  still  too 
weak  to  break 
up  any  of  the 
existing  molecu- 
lar groups,  the 
molecular  mag- 
nets turn  slightly 
against  the  mu- 
tual magnetic 
actions  of  the 

surrounding  molecules,  much  as  a  solid  body  which  is  not  dis- 
torted beyond  its  elastic  limits  yields  under  the  influence  of  an 
external  force.  But  as  the  magnetizing  force  c/if  increases,  a  point 
is  reached  at  which  the  molecular  groups  begin  to  break  up,  and 
the  molecules  to  set  themselves  parallel  to  the  direction  of  the 
field  &C.  The  part  of  the  curve  between  c  and  d  represents  the  cor- 
responding values  of  d  and  cK  when  this  breaking  down  of  the  mo- 
lecular groups  is  taking  place  most  rapidly.  It  will  be  seen  that 
within  this  region  a  very  small  increase  in  di  produces  a  very  large 
.increase  in  the  magnetization.  As  cK  is  still  further  increased  the 


=  Field-Intensity 
FIG.  122 


MAGNETIC   INDUCTION   IN   IKON  163 

magnetization  enters  upon  its  third  stage  represented  by  the  por- 
tion de  of  the  curve.  In  this  stage  the  molecular  groups  are  all 
broken  up,  but  the  molecules  have  not  yet  all  assumed  exact  par- 
allelism with  the  field.  As  c7f  still  further  increases,  the  molecules 
gradually  assume  the  exact  direction  of  the  field,  and  the  iron  is 
then  said  to  be  magnetically  saturated,  since  any  further  increase 
in  cTf  produces  no  increase  in  &. 

108.  Origin  of  the  factor  4?r.  The  reason  that  the  magnetiza- 
tion 3  is  not  taken  as  numerically  equal  to  the  number  of  lines 
due  to  the  iron,  namely  cB  —  cK ,  but  rather  to  (oB  —  e/f  )/4  TT,  is  as 
follows :  Since,  by  definition,  unit  magnetic  pole,  when  placed  at 
the  center  of  a  sphere  of  unit  radius,  produces  a  magnetic  field  of 
unit  strength  at  every  point  on  the  surface  of  the  sphere,  and  since 
a  field  of  unit  strength  is  represented  by  one  line  of  force  per 
square  centimeter,  it  is  clear  that  one  line  of  force  from  the  pole 
must  be  thought  of  as  piercing  each  of  the  4  TT  sq.  cm.  on  this 
unit  sphere.  Hence  we  must  imagine  4  TT  lines  of  force  as  emanat- 
ing from  every  unit  N  pole,  and  4  Trm  lines  of  force  as  emanating 
from  any  N  pole  which  contains  m  units  of  magnetism.  Now  we 
imagine  that  in  an  ideal  magnet  which  is  fully  saturated  there 
are  no  free  poles  except  at  the  very  ends  of  the  magnet,  so  that 
all  of  the  lines  which  are  associated  with  the  north  pole  emerge 
from  the  face  itself,  pass  around  in  closed  curves  to  the  S  face,  and 
then  return  through  the  magnet  to  the  j'Vface  (see  Fig.  101,  p.  135). 
If  the  strength  of  the  magnet's  poles  is  m,  then,  as  has  just  been 
shown,  the  number  of  these  lines  is  4  Trm.  The  number  of  unit 
poles  in  one  square  centimeter  of  the  face  is  then  not  equal  to  the 
number  of  lines  which  pass  out  of  this  face,  but  is  rather  equal  to 
this  number  divided  by  4  TT.  Now  it  has  been  decided  to  regard 
magnetization  as  the  number  of  unit  poles  per  square  centimeter, 
rather  than  as  the  number  of  lines  per  square  centimeter.  Hence 
it  was  that  we  divided  the  number  of  lines  per  square  centimeter 
due  to  the  iron,  namely  cB  —  cK ,  by  4  TT  in  order  to  obtain  the 
numerical  value  of  3  inside  the  ring.  Even  in  an  ordinary  unsatu- 
rated  magnet  in  which  lines  emerge  all  along  the  sides,  the  inten- 
sity of  magnetization  at  any  point  within  the  iron  is  defined  as 
the  number  of  lines  per  square  centimeter  there  present  divided 


164 


ELECTRICITY,  SOUND,  AND  LIGHT 


by  4  TT.  That  is,  it  is  the  number  of  units  of  molecular  magnets 
which  we  must  imagine  to  be  present  in  this  particular  square 
centimeter  in  order  to  account  for  the  number  of  lines  found  in 
this  square  centimeter  because  of  the  magnetization  of  the  iron. 

109.  Hysteresis.  The  curve  Icde  (Fig.  122),  representing  the 
values  of  the  magnetization  corresponding  to  a  magnetic  field  of 
constantly  increasing  magnetization,  is  shown  again  in  Figure  123. 
As  now  the  magnetic  field  is  allowed  to  decrease  to  its  original 
value  of  zero,  the  magnetization  assumes  successive  values  repre- 
sented by  the  curved  portion  ef.  The  explanation,  in  terms  of  the 


FIG.  123 

molecular  theory,  of  the  fact  shown  by  the  curve,  that  the  succes- 
sive values  of  the  decreasing  magnetization  do  not  lie  on  the  origi- 
nal curve  of  increasing  values,  is  as  follows :  When  the  magnetizing 
field  &C  is  withdrawn  or  diminished,  some  of  the  molecules  return 
under  their  mutual  actions  to  their  original  groupings,  but  many 
of  them  retain  their  alignment  until  this  is  broken  up  by  jars  or 
other  outside  forces.  In  hard  steel  the  difficulty  which  the  mole- 
cules experience  in  moving  out  of  any  positions  which  they  have 


MAGNETIC  INDUCTION  IN  IKON  165 

once  assumed  is  very  large,  while  in  soft  iron  it  is  small.  In  general, 
then,  the  magnetization  is  said  to  "  lag  behind  "  the  field-intensity 
which  induces  it.  This  phenomenon,  known  as  hysteresis,  is  an 
important  characteristic  of  all  magnetic  substances. 

When  the  field  has  been  decreased  to  zero  the  value  of  the  mag- 
netization retained  by  the  iron  is  known  as  the  residual  magneti- 
zation. In  practice,  however,  the  knowledge  of  this  quantity  is  of 
little  value.  It .  is  in  general  more  desirable  to  know  the  ratio 
which  the  residual  magnetization,  represented  by  the  ordinate  5/, 
bears  to  the  maximum  value  of  the  induced  magnetization,  repre- 
sented by  the  ordinate  en.  This  ratio  is  defined  as  the  retentivity 
and  is  usually  stated  in  per  cent. 

In  order  that  the  magnetization  shall  be  again  zero,  the  direc- 
tion of  the  field  must  be  reversed,  and  the  value  of  its  intensity 
caused  to  increase  in  this  opposite  direction.  Under  the  action 
of  this  reversed  field,  the  values  of  which  will  for  convenience 
be  called  negative  and  plotted  to  the  left  of  the  vertical  axis,  the 
value  of  the  magnetization  rapidly  decreases  in  a  manner  repre- 
sented by  the  curve  fg.  The  value  of  the  negative  intensity  which 
must  be  applied  to  reduce  the  residual  magnetization  to  zero  is 
known  as  the  coercive  force.  Numerically,  it  is  represented  by  the 
abscissa  bg. 

It  is  evident  that  if  this  oppositely  directed  field  is  still  further 
increased  in  intensity,  there  must  result  a  magnetization  opposite 
in  direction  to  the  original  magnetization.  Now  let  the  maximum 
value  of  the  original  intensity,  shown  in  the  figure  by  the  abscissa 
bn,  be  represented  by  the  symbol  +<$fm.  It  is  of  interest  to  consider 
the  effect  of  hysteresis  upon  the  successive  values  of  the  magneti- 
zation corresponding  to  a  cyclic  change  of  c?f,  in  which  cK  passes 
continuously  through  all  the  values  from  +<2fm  to  —3im  and  back 
again  to  +&Cm.  After  a  sufficient  number  of  repetitions  of  this 
cycle,  the  values  of  3  are  found  to  lie  on  a  closed  curve  of  the  gen- 
eral form  represented  by  efghije  and  known  as  the  hysteresis  loop.* 

*  In  the  commercial  transformers  described  in  section  101,  page  153,  the 
iron  core  undergoes  such  cyclic  changes  in  its  magnetic  state.  Work  is  done 
in  producing  these  changes,  and  this  loss  of  energy  due  to  hysteresis  is  of 
practical  importance  in  the  design  of  transformers. 


166  ELECTRICITY,  SOUND,  AND  LIGHT 

110.  Formula  for  field  strength  within  a  long  solenoid.    It  is 

generally  desirable  in  experiments  on  cB  and  3K  to  determine  cB 
by  a  test-coil  method,  applying  the  principles  of  section  105  in  a 
manner  which  will  be  explained  in  the  next  section  ;  but  instead 
of  using  a  wooden  ring  for  the  determination  of  ef{  it  is  customary 
to  calculate  this  quantity  from  the  number  of  turns  z  per  centi- 
meter of  length  of  the  solenoid  and  the  current  /  flowing  about 
it.  The  relation  between  these  quantities  is  as  follows  : 

v  _  4  TTZl 

~~' 


The  origin  of  this  formula 
may  be  seen  from  the  follow- 
ing considerations. 

'  The  force  which  a  unit  mag- 

FIG.  124 

net  pole  placed  at  a  point  p 

(Fig.  124)  on  the  axis  of  a  long  solenoid  experiences  because  of  a 
current  /  flowing  around  it  is,  of  course,  simply  the  resultant  of  all 
the  magnetic  forces  exerted  upon  it  because  of  the  magnetic  fields 
surrounding  all  elements  of  the  wire  constituting  the  solenoid. 
Consider  first  the  value  of  the  force  exerted  upon  the  pole  at  p 
by  the  elements  of  current  which  lie  on  a  small  element  of  area 
A/As  of  the  surface  of  the  solenoid.  If  there  are  z  turns  per  centi- 
meter, there  will  be  zbl  elements  of  current  in  the  distance  A/. 
Each  of  these  has  a  length  As.  The  force  which  each  element 
exerts  on  a  unit  magnet  pole  at  the  point  p  a  distance  r  from  the 

element  is,  by  definition  of  unit  current  (see  sect.  30,  p.  36),  —  ^-  >  and 
the  total  force  /  due  to  all  the  elements  z&l  is  given  by 


Since,  from  considerations  of  symmetry,  it  is  obvious  that  the 
resultant  force  exerted  on  the  pole  at  p  by  the  whole  solenoid  must 
be  parallel  to  the  axis  of  the  solenoid,  we  are  here  concerned  only 
with  the  component  of  /  in  the  direction  of  the  axis.  Calling 
this  /',  we  have  i^lAscosff  _ 

J  '2  \     ' 


MAGNETIC  INDUCTION  IN  IRON  167 

But  precisely  as  in  section  73,  page  108,  —  -  is  the  solid  angle 

subtended  at  p  by  the  elementary  area  A/As.  Hence,  representing 
this  solid  angle  by  u,  we  have 

f  =  Izu.  (5) 

That  is,  the  magnetic  force  exerted  parallel  to  the  axis  by  any  ele- 
ment of  the  solenoid  is  the  solid  angle  subtended  at  the  point  by  the 
element,  multiplied  by  the  current  and  by  the  number  of  turns  per 
centimeter  of  length  of  the  solenoid.*  The  total  force  3i  acting  upon 
the  unit  pole  at  p  is  the  sum  of  the  forces  due  to  all  the  elements 
of  the  surface,  i.e.  di  =  2lzu.  Since  /  and  z  are  the  same  for  all 
elements,  we  may  write  this  in  the  form  cK  =  Iz2u,  and  if  the 
solenoid  is  so  long  that  we  may  neglect  the  solid  angle  subtended 
by  its  open  ends  in  comparison  with  the  solid  angle  subtended  by 
its  surface,  we  have,  since  the  solid  angle  about  a  point  is  4  TT, 

W  =  4  -irzl.  (6) 

If  /  is  expressed  in  absolute  units  of  current,  the  field  strength  Si 
will  be  expressed  in  gausses.  If  /  is  in  amperes,  in  order  to  obtain 
di  in  gausses  we  must  write 

«-*? 

Since  this  analysis  holds  for  any  point  within  the  solenoid  which 
is -not  too  close  to  its  ends,  it  is  obvious  that  the  field  strength 
within  a  long  solenoid  is  uniform.  If  the  solenoid  is  a  ring,  the 
deduction  is  rigorously  correct  for  all  points  within  the  ring.  If 
it  is  a  cylinder,  the  general  rule  is  that  the  length  must  be  as  much 
as  twelve  times  the  diameter  in  order  that  equation  (7)  may  be 
applied  without  appreciable  error  to  find  <?f  at  the  center.  In  order 
to  see  the  reason  for  this  rule  it  is  only  necessary  to  consider  for 
what  ratio  of  length  to  diameter  the  solid  angle  subtended  at  the 
center  by  the  ends  becomes  negligible  in  comparison  with  the  solid 
angle  subtended  by  the  surface  of  the  solenoid. 

*  Although  this  conclusion  has  been  arrived  at  by  considering  a  point  on  the 
Oicis,  it  holds  for  all  points  within  the  solenoid,  as  can  be  shown  by  resolving 
each  element  of  current  into  two  components,  one  parallel  and  one  perpendicular 
to  the  line  connecting  the  point  and  the  element. 


168 


ELECTEICITY,  SOUND,  AND  LIGHT 


111.  Ballistic  method  of  determining  the  magnetization  curve. 
The  method  to  be  described  for  plotting  the  curve  of  magnetiza- 
tion discussed  in  section  107  consists,  first,  in  finding  a  set  of 
corresponding  values  of  cS  and  eTf,  and  then,  from  the  relation 
c/  =  (oB  —  &C)/±  TT,  finding  the  values  of  3  corresponding  to  each 
value  of  cTf .  The  material  to  be  examined  is  given  the  form  of  a 
ring  about  which  is  wound  uniformly  a  coil  of  wire  represented 
by  P  in  Figure  125.  In  series  with  the  coilP  there  is  an  ammeter 
A  and  a  resistance  11'  which  may  be  varied  without  causing  an 
interruption  of  the  current  passing  through  it.  A  commutator  c 
admits  of  a  reversal  of  the  current  in  the  coil.  At  some  point  of 
the  coil  P  is  wound  a  small  test  coil  S  which  is  connected  through 
a  resistance  R  to  a  ballistic  galvanometer  G.  Starting  with  zero 


FIG.  125 

current  in  the  coil  P,  the  current  is  increased  by  a  series  of  small 
steps  by  adding  one  by  one  the  conductors  joined  in  parallel  in  R'. 
The  value  of  Si  corresponding  to  each  value  of  the  current  is  then 
calculated  from  equation  (7).  Corresponding  to  these  changes  in 
the  value  of  cK  there  are  changes  in  the  induction  cB.  Each 
change  in  induction,  which  will  be  represented  by  Ac$,  is  propor- 
tional to  the  galvanometer  throw  which  it  produces,  and  might  be 
measured  by  the  relations  given  in  equation  (1),  section  102,  but 
is  most  easily  found  by  inserting  an  earth  inductor  El  into  the 
galvanometer  circuit  and  comparing  the  throw  produced  by  it  as  it 
rotates  in  the  known  field  of  the  earth  with  the  throws  produced 
by  A  08.  Obviously  the  total  induction  cB  in  the  iron  at  any  time 


MAGNETIC  INDUCTION  IN  IRON 


169 


is  the  algebraic  sum  of  all  the  preceding  values  of  AcB.  Corre- 
sponding values  of  cB  and  oft  are  then  plotted.  The  magnetization 
curve,  i.e.  the  curve  connecting  3  and  oK,  may  be  found  from  these 
values  by  the  method  of  the  next  section. 

112.  Graphical  transformation  of  SB  and  &C  curve  to  an  3  and 
ofC  curve.  The  number  of  lines  representing  the  induction  cB  is  in 
general  so  many  times  greater  than  the  number  representing  the 


16000 
14000 

_-=»-  

—  •- 

'200. 
LLOJL 

30Q_ 
QOQ_ 

600_ 
400. 

/oo 

^ 

_  

' 

12000 

/ 

^ 

A 

10000 

, 

8000 





/ 

i 

6000 

i 

1 

4000 





-- 

\ 

ZOOO 





i 













_ 

.      



/ 

s_ 

3       4       3       6        7   '    &       9       10      II       12 

FIG.  126 

corresponding  value  of  cTf  that  they  are  plotted  on  very  different 
scales.  Thus  in  Figure  126  is  shown  the  relation  of  oB  and  3i  for 
increasing  values  of  3i.  The  scale  of  4B  is  about  one  thousandth 
of  that  for  ott '.  To  find  the  value  of  3  corresponding  to  any  value 
of  c?f  it  is  necessary  to  subtract  c?f  from  cB  and  divide  by  4  TT  ;  that 
is,  cf  =  (cB  —  c?T)/4  TT.  For  all  practical  purposes,  however,  in  dealing 


170  ELECTRICITY,  SOUND,  AND  LIGHT 

with  values  of  3  below  saturation,  it  is  sufficient  to  neglect  3i  in 
comparison  with  eB  and  to  write  &  =  e8/47r.  Now,  if  the  scale  of 
the  axis  of  cS  in  Figure  126  is  changed  by  letting  each  division 
represent  the  1/4  TT  part  of  its  original  value,  the  division  of  cB  by 
4?r  is  at  once  performed  for  all  values  of  cB.  In  other  words,  within 
these  limits,  one  single  curve  represents  both  the  relations  of  cB  and 
&C  and  of  3  and  3i '.  The  new  scale  for  the  axis  of  ordinates  is  shown 
at  the  right  of  the  scale  for  cB.  The  value  of  $  corresponding  to 
any  value  of  3{  is  then  read  at  once  from  the  curve  on  this  new 
scale.  This  method  of  transforming  a  cB  and  3i  curve  to  an  3  and 
cTf  curve  may  of  course  be  applied  to  the  entire  hysteresis  loop. 

113.  Demagnetization  of  the  specimen  to  be  tested.    As  is  evi- 
dent from  the  preceding  discussion  of  hysteresis,  the  fact  that  the 
magnetizing  force  &C  is  zero  is  no  evidence  that  the  magnetization 
3  is  also  zero.    And  further,  even  though  both  be  zero,  the  pre- 
vious magnetic  history  of  the  specimen  under  examination  may 
be  such  that  it  has  a  distinct  set  or  tendency  toward  a  magneti- 
zation in  some  particular  direction.    Now  it  is  of  practical  impor- 
tance to  compare  the  magnetic  qualities  of  various  substances  by 
their  magnetization  curves.    In  order  that  such  a  comparison  may 
be  made,  it  is  evident  that  the  original  state  of  the  specimens  must 
be  the  same  at  the  time  the  observations  are  begun.    This  is  made 
possible  by  demagnetizing  as  follows :  starting  with  a  value  of  the 
current  in  the  coil  P  (Fig.  125)  such  that  the  value  of  <?f  is  higher 
than  any  value  to  which  the  specimen  has  been  recently  subjected, 
the  current  is  gradually  decreased  to  zero.    During  this  decrease 
the  direction  of  the  current  is  made  to  undergo  a  series  of  rapid 
alternations  produced  at  the  commutator.   As  a  result  of  this  oper- 
ation the  substance  shows  no  predisposition  toward  a  magnetiza- 
tion in  any  one  direction. 

114.  The  magnetometric  method  of  testing  magnetic  substances. 
If  the  specimen  to  be  tested  is  in  the  form  of  a  bar,  it  may  be  mag- 
netized by  placing  it  inside  a  long  solenoid.     The  magnetization, 
which  is  the  pole  strength  per  unit  area  of  cross  section,  may  then 
be  found  by  the  use  of  the  magnetometer  as  shown  in  Chapter  II. 
The  chief  objection  to  this  method  lies  in  the  fact  that  the  free 
magnetic  poles  formed  near  the  ends  of  the  bar  set  up  within  the 


MAGNETIC  INDUCTION  IN  IKON  171 

solenoid  a  field  in  the  opposite  direction  to  the  magnetizing  field 
off,  and  thus  tend  to  weaken  this  field.  The  actual  amount  of  the 
weakening  at  any  instant  depends  upon  the  pole  strength  of  the 
bar  at  that  instant  and  upon  its  form.  It  may  be  calculated  by 
an  analysis  which  is,  however,  beyond  the  scope  of  this  text. 

The  ballistic  method  is  free  from  this  objection,  for  in  a  closed 
ring  there  are  no  free  poles.  But  it  is  open  to  the  objection  that 
it  takes  into  account  only  sudden  changes  in  the  induction.  The 
growth  of  the  magnetization  is,  however,  subject  to  a  small  gradual 
increase,  which  continues  for  some  time  after  a  sudden  change  in 
the  magnetizing  force.  This  phenomenon  is  known  as  "  magnetic 
creeping." 

115.  Permeability  and  susceptibility.  In  the  study  of  the  mag- 
netic properties  of  various  materials,  two  quantities,  known  as  per- 
meability and  susceptibility,  are  of  particular  interest.  The  ratio 
of  the  induction  cB  to  the  corresponding  value  of  the  magnetizing 
force  off,  as  taken  from  the  ascending  curve  of  magnetization  for  a 
specimen  previously  demagnetized,  is  defined  as  the  permeability 
of  the  specimen  for  that  value  of  cB  or  ${.  That  is,  representing 
permeability  by  ft,  we  have 

-I- 

In  the  same  manner  the  ratio  of  the  magnetization  c7,  taken 
from  the  ascending  curve  of  magnetization  for  a  specimen  pre- 
viously demagnetized,  to  the  corresponding  value  of  the  magnetiz- 
ing force  &C,  is  known  as  the  susceptibility.  Hence,  representing 
susceptibility  by  K,  we  have 

•=!• 

And  since  cB  =  &C  +  4  ird,  we  have 

fji#C=#C  +  4;7TK&C,  (10) 

Or  fJL  =  l+  4:7TK.  (11) 

The  permeability  of  a  specimen  of  soft  iron  attains  a  maximum 
of  about  2500  at  some  point  of  the  second  stage  of  magnetization 


172 


ELECTKICITY,  SOUND,  AND  LIGHT 


referred  to  in  section  107,  page  161.  As  the  values  of  the  induo 
tion  are  increased  still  further  the  permeability  decreases.  As  sat- 
uration is  approached 
(<$  =  16,000)  the  value 
of  /x  is  very  low,  only 
about  500.  For  very 
high  values  of  the  in- 
duction, for  example, 
<®= 4 5,000  (which  cor- 
responds to  3i=  24,000), 
ft  is  reduced  nearly  to 
one.  This  is  what  we 
should  expect;  for 

beyond    saturation   &C 
FIG.  127  '  , 

becomes  large  as  com- 
pared to  8,  which  remains  practically  constant,  and  4  TTK  in  equa- 
tion (11)  approaches  zero.  A  curve  showing  the  relation  between 
/-i  and  cB  for  a  specimen  of  soft  iron  is  shown  in  Figure  127. 


EXPERIMENT  15 

Object.  To  plot  the  curve  of  magnetization  for  a  sample  of  iron ;  also  to 
plot  the  hysteresis  loop  for  the  same  sample. 

Directions.  The  sample  to  be  tested  is  in  the  form  of  a  ring  (Fig.  128), 
the  mean  peripheral  length  of  which  is  some  25  or  30  times  the  diameter 
of  its  cross-sectional  area.  The  ring  is  wound  uniformly  in  two  layers 
with  about  20  turns  of  wire  per  cen- 
timeter of  mean  peripheral  length. 
The  layers  have  separate  terminals 
and  are  to  be  connected  in  series. 
Connect  as  in  Figure  125.  The  mil- 
liamrneter  A  is  one  of  range  0  to 
500.  The  resistance  R'  is  a  specially 
constructed  set  of  resistances  all 
in  parallel  and  controlled  by  indi- 
vidual knife  switches  (see  Fig.  129). 

The  battery  B  should  contain  a  number  of  cells  such  that  with  all  these 
resistances  in  the  circuit  the  current  is  about  .490  ampere.  The  commu- 
tator c  is  a  specially  constructed  rotary  commutator  shown  in  Figure  130. 


FIG.  128 


MAGNETIC  INDUCTION  IN  IRON 


173 


The  terminals  of  the  battery  circuit  are  to  be  connected  to  the  binding 
posts  e  and  /,  those  of  the  coil  to  y  and  h. 

The  test  coil,  consisting  of  a  number  of  turns  of  fine  wire,  is  wound  over 
the  primary,  covering  a  space  of  5  to  10  mm.  of  length.  In  series  with 
this  there  is  an  earth  in- 
ductor El  (Fig.  125),  a 
resistance  R  (about  20,000 
ohms),  and  the  ballistic  gal- 
vanometer G. 

Open  the  galvanometer 
circuit  and  demagnetize  the  FJG    12Q 

iron  ring  as  follows.    Start- 
ing with  all  the  resistances  in,  rotate  the  commutator  rapidly  and  one  by 
one  open  the  resistance  switches,  opening  the  highest  resistance  last. 

Now  connect  the  galvanometer  circuit.  Close  the  switches  one  by  one 
in  the  opposite  order  to  that  in  which  they  were  opened,  observing  as  each 
switch  is  closed  the  galvanometer  deflection  caused  and  the  value  of  the 
current.  This  group  of  observations  constitutes  the  data  for  the  ascending 
cB  and  cK  curve. 

Without  altering  any  connections',  obtain  the  data  for  the  hysteresis 
loop  as  follows.  Open  the  switches,  one  at  a  time,  and  observe  the  cor- 
responding deflections  and  currents.  When  they  .are  all  open  turn  the 
commutator  carefully  so  as  to  reverse  the  direction  of  the  current  through 

the  primary,  then  increase  it  to  its 
maximum  value  by  successively  clos- 
ing, and  afterward  decrease  it  by 
successively  opening,  the  switches,  ob- 
serving the  corresponding  deflections 
and  current  values  as  before.  Call 
the  deflections  of  the  galvanometer 
in  the  original  direction  plus,  and  in 
the  opposite  direction  minus.  When 
the  current  has  been  brought  to  zero 
reverse  again  the  direction  and  ob- 
tain a  set  of  readings  for  the  current 
increasing  in  the  original  direction. 

Without  altering  any  connections, 
take  a  set  of  three  readings  for  the 
throw  dr  produced  by  the  earth  in- 
ductor in  cutting  the  horizontal  com- 
ponent of  the  earth's  magnetic  field. 

Plot  the  values  of  current  and  the  algebraic  sum  of  the  galvanometer 
throws  as  abscissas  and  ordinates  respectively  on  a  sheet  of  coordinate 
paper.  The  curve  thus  found  gives  the  relation  between  the  magnetizing 


FIG.  130 


174  ELECTRICITY,  SOUND,  AND  LIGHT 

current  /  in  amperes  and  the  galvanometer  throw  due  to  the  magnetic 
induction  thus  caused.*  Since  SK  is  proportional  to  this  current  and  $  pro- 
portional to  the  throw,  this  curve  may  at  once  be  transformed  into  a  cB 

and  (5f  curve  by  changing  the  scale.    Thus  from  the  relation  5f  =  —  ^—  find 

the  value  of  /  which  corresponds  to  a  value  of  5f  =  10.  Through  points  on 
the  axis  of  current-  values  that  correspond  to  the  plus  and  minus  of  this 
value  of  the  current  draw  vertical  lines  in  red  ink.  With  a  scale,  or  a  pair  of 
hairspring  dividers,  divide  the  space  between  each  line  for  which  &C  =  10  and 
the  axis,  or  line  for  which  SH  =  0,  into  ten  equal  parts.  Lay  off  on  either  side 
of  these  two  vertical  lines  other  points  corresponding  to  5f  =  11,  #f  =  12, 
etc.  Through  all  these  points  draw  vertical  lines  in  red  ink  and  number 
accordingly. 

Similarly,  the  transformation  of  the  vertical  axis  into  an  axis  giving 
values  of  cB  may  be  made.  Thus  if  //  represents  the  value  of  the  horizontal 
component  of  the  earth's  field  at  the  point  where  the  earth  inductor  is 
placed,  A'  the  total  area  of  the  inductor,  and  df  the  average  throw  of  the 
galvanometer  when  the  inductor  is  rotated,  and  if  n  is  the  number  of  turns 
on  the  test  coil,  a  the  average  area  of  the  iron  ring  (for  all  the  lines  pass 
through  it),  and  d  the  total  throw  -which  the  galvanometer  would  make 
because  of  the  insertion  of  c&  lines  per  square  centimeter  through  the  iron 
ring,  then,  since  the  throws  are  proportional  to  the  change  in  the  number  of 
lines,  we  have  d  '  ^  A'H 


Solve  this  relation  to  find  the  value  of  the  galvanometer  throw  d  which 
corresponds  to  a  change  of  induction  cB  of  10,000.  Call  the  point  on  the 
vertical  axis  which  corresponds  to  this  value  of  the  deflection  10,000,  and 
divide  the  space  between  this  point  and  the  horizontal  axis  into  ten  equal 
parts  as  before.  Continue  this  division  to  either  side  of  the  points  repre- 
senting $  =  +  10,000  and  <0=  -  10,000  in  order  to  obtain  &=  11,000,  etc., 
and  rule  through  these  divisions  horizontal  red  ink  lines.  Reading  on  the 
scale  represented  by  the  red  lines,  we  now  have  the  cB  and  <K  curve  for  the 
specimen. 

Plot  on  the  same  sheet  and  draw  the  curve  in  red  ink,  showing  the 
relation  /u,=o3/c?f,  as  found  from  the  ascending  curve  of  magnetization. 
Use  the  axis  of  S3  already  drawn  as  one  of  the  axes  of  this  new  curve. 
Tabulate  the  values  used  in  plotting  the  cB  and  c/f  curve  in  one  corner  of 
the  sheet.  Below  these  place  the  value  of  the  retentivity  and  of  the 
coercive  force  as  found  from  the  curve  after  the  manner  of  section  109. 

*  If  the  hysteresis  loop  is  not  found  to  close  completely  at  the  upper  end,  it  is 
because  the  iron  has  not  been  carried  through  this  particular  cycle  a  sufficient 
number  of  times  to  lose  previous  tendencies  which  were  not  entirely  eliminated 
by  demagnetizing. 


MAGNETIC   INDUCTION  IN  IRON  175 


EXAMPLE 

The  coil  surrounding  the  ring  had  a  total  of  579  turns.      The  average 
radius  of  the  ring  was  4.75  cm.    Hence 

579  579         I 

and         SK  =  4  TT 


27r4.75  271-4.75    10 

Hence  a  current  of  .4102  amperes  produced  a  field  of  eft  =  10  gausses. 
The  test  coil  consisted  of  80  turns.  The  area  a  through  which  the  flux 
passed  was  that  of  the  iron  core,  since  all  the  lines  were  in  the  iron  ring. 
The  average  diameter  of  the  ring  wras  1.434cm.  Hence  the  area  was 
TT  x  .7172sq.  cm.  The  eartli  inductor  had  a  total  area  found  by  multiplying 
600,  the  number  of  turns,  by  208.7,  the  average  area.  The  mean  throw 
produced  by  the  earth  inductor  was  1.40cm.  The  value  of  H  as  found  in 
Experiment  13  was  .1845  gausses.  Hence  a  deflection  of  d  =  39.14cm.  on 
the  scale  corresponded  to  a  value  of  cB=  10,000  (see  eq.  12). 

The  curves  plotted  were  of  the  form  of  those  in  Figures  123  and  126. 
The  maximum  value  of  /  used  was  .495  ampere.  This  corresponded  to  a 
value  of  ffC  of  12.05  gausses,  and  a  value  of  cB  of  16,300  lines.  The  coercive 
force  was  found  to  be  1.5  gausses  and  the  retentivity  61  per  cent.  The 
permeability  reached  a  maximum  value  of  2470  when  the  magnetizing 
force  SK  was  2  gausses. 


CHAPTER   XVI 
ELECTROLYTIC  CONDUCTION 

116.  Early  views  of  electrolytic  conduction.  Faraday  and  his 
immediate  successors  regarded  the  decomposition  which  attends 
the  passage  of  a  current  through  an  electrolyte*  as  an  immediate 
result  of  the  electric  current.  Thus,  when  hydrochloric  acid  (HC1) 
was  dissolved  in  water,  they  imagined  the  acid  molecules  to  remain 
intact  in  the  solution,  each  molecule  consisting  of  a  positively 
charged  atom  of  hydrogen  (H)  and  a  negatively  charged  atom  of 
chlorine  (Cl)  which  clung  together  under  the  influence  of  their 
mutual  attractions.  When  the  positively  and  negatively  charged 
terminals  of  a  battery  were  immersed  in  the  solution  the  molecules 
were  supposed  to  be  split  up,  by  the  influence  of  the  charges  on  the 
battery  terminals,  into  positive  hydrogen  ions  and  negative  chlorine 
ions.  These  ions  migrated  through  the  solution  in  opposite  direc- 
tions to  the  two  opposite  electrodes.  As  the  H  ions  collected  about 
the  negative  electrode  some  of  them  touched  it  and  gave  up  their 
positive  charges.  They  received  in  return  negative  charges  and 
were  then  in  condition  to  unite  with  the  free  H  ions  which  had 
not  yet  lost  their  positive  charges,  and  thus  to  form  neutral  mole- 
cules of  hydrogen  gas  (H2),  which  collected  as  bubbles  on  the 
plate  or  rose  to  the  air  above,  f 

The  mechanism  by  which  the  ions  passed  through  the  solution 
was  supposed  to  be  as  follows.  Under  the  influence  of  the  electric 

*  The  student  may  well  reread  in  this  connection  sections  24  and  31. 

t  In  terms  of  the  electron  theory  this  would  mean  that  the  positive  H  ion, 
which  is  positive  because  it  lacks  one  electron,  receives  one  from  the  electrode 
in  giving  up  its  charge,  and  then  one  in  excess  by  virtue  of  which  it  is  repelled 
from  the  plate,  and  is  put  into  condition  to  unite  with  another  positively 
charged  II  ion  to  form  a  neutral  molecule  of  hydrogen.  The  result  would  of 
course  be  precisely  the  same  if  all  of  the  H  ions  gave  up  their  positive  charges 
to  the  plate  and  then  in  some  not  understood  way  united  to  form  hydrogen 
molecules. 

176 


7 
-  -f 


oo   60   66   06   66 


ELECTROLYTIC   CONDUCTION  177 

field  due  to  the  electrodes  the  HC1  molecules  first  oriented  them- 
selves as  in  the  figure  (Fig.  131),  the  +  half  turning  toward  the 

—  electrode  and  the  —  half  toward  the  +  electrode.    When  the  field 
became  strong  enough,   de- 
composition took  place ;  the 

-h  ion  of  molecule  5  went  to 
the  —  electrode,  while    the 

—  ion   of  5  at  once  united 
with  the  +  ion   of   4,   the 

—  of  4   with   the  +   of  3, 
and   so  on  throughout  the 

entire  chain.  Thus  the  current  was  transferred  by  virtue  of  the 
continual  exchange  of  partners  taking  place  in  the  HC1  mole- 
cules in  the  solution.  This  picture  is  due  to  Grotthus  (1805), 
and  is  known  as  "  the  Grotthus  chain  theory "  of  electrolytic 
conduction. 

117.  Clausius's  dissociation  hypothesis.  The  first  serious  objec- 
tions which  were  raised  to  this  Grotthus  chain  theory  were  brought 
forward  by  the  German  physicist  Clausius  in  1857.  The  chief 
argument  which,  he  advanced  against  the  point  of  view  of  Grotthus 
was  as  follows.  The  view  demanded  that  a  certain  definite  P.D. 
exist  between  the  electrodes  before  any  decomposition  could  occur, 
and  therefore  before  any  current  whatever  could  flow.  As  soon  as 
this  critical  P.D.  has  been  exceeded  the  current  should  rise  instantly 
to  a  considerable  value.  As  a  matter  of  fact,  the  current  through 
an  electrolyte  does  not  behave  in  this  manner.  It  is  true  that  it 
does  require  a  certain  critical  P.D.  to  produce  a  continuous  decom- 
position of  an  electrolyte  if  the  electrodes  are  of  different  material 
from  that  deposited  on  the  cathode,*  for  then  a  back  E.M.F.  of 
polarization  is  introduced  because  of  the  formation  of  essentially 
new  plates  by  the  deposit ;  and  this  back  E.M.F.  must  be  exceeded 
before  any  continuous  separation  of  the  elements  in  the  solution 
can  take  place.  If,  however,  the  electrodes  are  of  the  same  nature 
as  the  metallic  ion  in  solution,  as  when  copper  electrodes  are 
dipped  into  a  copper  sulphate  solution,  then  the  current  which 

*  See  section  75,  page  113. 


178  ELECTRICITY,  SOUND,  AND  LIGHT 

flows  is  directly  proportional  to  the  RD.  applied.  In  other  words, 
Ohm's  law  holds  for  such  a  solution  as  well  as  for  metallic  con- 
ductors, and  there  is  no  critical  P.D.  below  which  the  current 
is  zero. 

In  order  to  account  for  this  fact,  Clausius  suggested  the  follow- 
ing modification  of  the  theory  of  electrolytic  conduction.  He  held 
that  in  every  electrolyte  the  molecules  of  the  substance  in  solu- 
tion are  already  dissociated,  in  part  at  least,  into  their  electrically 
charged  constituents.  The  mechanism  by  which  this  dissociation 
was  effected  might  be  left  unsettled.  Clausius's  own  view  was  that 
in  the  impacts  which  the  molecules  make  with  the  water  mole- 
cules an  occasional  collision  is  so  violent  as  to  split  up  the  molecule 
into  its  constituents.  Each  of  these  electrically  charged  parti- 
cles then  moves  about  among  the  water  molecules  until  it  meets 
another  particle  produced  by  dissociation,  but  of  opposite  kind, 
when  it  unites  with  it  to  form  a  new  molecule  of  the  compound. 
A  condition  of  so-called  active  equilibrium  is  then  set  up  when 
the  number  of  molecules  which  become  dissociated  per  second  is 
equal  to  the  number  of  new  molecules  formed  per  second  by 
recombination. 

According  to  this  view  at  any  instant  in  any  conducting  solu- 
tion a  fraction  of  the  molecules  is  dissociated  into  its  ions,  and 
these  free  ions  begin  to  move  toward  the  electrodes  the  moment 
a  P.D.  is  applied.  The  current  does  not  produce  the  dissociation, 
but  rather  the  dissociation  is  the  necessary  condition  for  the  pas- 
sage of  a  current.  Substances  which  do  not  in  part  dissociate 
spontaneously  upon  going  into  solution  cannot  conduct.  This  view 
accounts  for  the  fact  that  a  critical  value  of  the  electrical  field 
established  between  the  electrodes  is  not  required  to  start  a 
current,  for  according  to  it  the  field,  instead  of  having  to  pull 
the  ions  apart,  has  merely  to  set  those  which  are  already  sepa- 
rated into  motion  toward  the  appropriate  electrodes.  Further- 
more, under  these  circumstances  Ohm's  law  would  be  expected 
to  hold,  for  doubling  the  applied  P.D.  would  merely  double  the 
speed  with  which  the  free  ions  move  toward  the  electrodes, 
and  hence  the  number  which  give  up  their  charges  per  second 
to  the  electrodes. 


ELECTROLYTIC   CONDUCTION  179 

118.  Osmotic  phenomena.  This  dissociation  theory  did  not 
receive  much  attention  until  1887  when  it  received  strong  sup- 
port from  an  unexpected  source,  namely,  the  epoch-making  work 
of  Van  't  Hoff  upon  osmotic  phenomena. 

The  osmotic  pressure  of  a  solution  is  most  directly  measured  by 
measuring  the  force  with  which  the  pure  solvent  (for  example, 
water)  tends  to  enter  the  solution  through  a  membrane  which  is 
permeable  to  the  solvent  but  not  to  the  dissolved  substance.  It  is 
most  easily  and  most  accurately  measured,  however,  by  observing 
the  number  of  degrees  the  freezing  point  of  the  given  solution  is 
depressed  below  the  freezing  point  of  the  pure  solvent.  Now  it 
was  discovered  that  while  the  osmotic  pressure  exerted  by  sub- 
stances in  solution  is  in  general  simply  proportional  to  the  num- 
ber of  molecules  dissolved  in  a  liter,  and  is  not  at  all  dependent 
upon  the  nature  of  the  dissolved  substance,  this  is  not  true  for 
electrolytes,  for  these,  in  general,  show  abnormally  high  values 
of  this  pressure.  Indeed,  the  osmotic  pressure  produced  by  intro- 
ducing a  given  number  of  molecules  of  NaCl  or  KC1  (for  exam- 
ple, one  gram-molecule*)  into  a  given  number  of  liters  of  water 
is,  in  dilute  solutions,  exactly  twice  as  great  as  the  osmotic  pres- 
sure produced  by  introducing  the  same  number  of  molecules  of 
sugar  into  the  same  number  of  liters  of  water.  The  most  natu- 
ral interpretation  of  this  phenomenon  is  that  each  of  the  mole- 
cules of  NaCl  or  KC1  breaks  up  into  two  parts  upon  going  into 
solution. 

When  it  is  found  that  practically  all  electrolytes  exhibit  this 
abnormal  osmotic  pressure  as  measured  by  the  depression  of  the 
freezing  point  per  gram-molecule  of  dissolved  substance ;  that  no 
substances  except  electrolytes  exhibit  abnormally  high  osmotic 
pressures;  that,  in  dilute  solution,  substances  like  NaCl  and  KC1, 
which  can  break  up  into  two  ions  only,  never  show  more  than 
double  the  normal  depression  as  observed  for  non electrolytes, 


*  A  "gram-molecule  "  of  any  substance  is  a  mass  of  the  substance  in  grams 
equal  to  its  molecular  weight.  The  advantage  in  comparing  gram-molecules 
instead  of  grams  of  different  substances  lies  in  the  fact  that  the  gram-molecule 
represents  for  all  substances  exactly  the  same  number  of  molecules.  A  solution 
containing  one  gram-molecule  per  liter  of  the  solution  is  called  a  normal  solution. 


180  ELECTRICITY,  SOUND,  AND  LIGHT 

while  substances  like  SrCl2  which  might  give  three  ions  (Sr,  Cl,  Cl) 
actually  do  show  between  two  and  three  times  the  normal  depres- 
sion, but  never  more  than  three  times,*  it  is  evident  that  Clausius's 
theory  of  dissociation  has  received  powerful  support.  Further- 
more, these  phenomena  seem  to  indicate  that  in  dilute  solutions 
the  dissociation  of  some  substances  is  practically  complete ;  that  is, 
that  all  of  the  molecules  of  the  solute  are  split  up  into  their  con- 
stituent ions.  Further  light  has  been  thrown  upon  this  subject  by 
a  study  of  so-called  "  molecular  conductivity." 

119.  Molecular  conductivity.  The  significance  of  the  term 
molecular  conductivity  may  be  seen  from  the  following  illustra- 
tion. Imagine  a  vessel  of  indefinite  height,  of  rectangular  horizon- 
tal cross  section,  and  provided  with  two  platinum  sides  1  cm.  apart. 
Let  a  normal  solution  of  some  electrolyte  be  placed  in  this  cell 
and  the  resistance  between  the  platinum  plates  measured.  The 
reciprocal  of  this  resistance  is  called  the  molecular  conductivity  at 
a  dilution  of  one.  Let  a  liter  of  pure  water  be  added  to  the  con- 
tents of  the  vessel  and  the  resistance  measured  again.  The  recipro- 
cal of  this  is  the  molecular  conductivity  at  a  dilution  of  two. 
Similarly  the  molecular  conductivity  at  any  dilution  may  be  found. 
It  is  evident  that  under  these  circumstances  a  comparison  is  made 
of  the  conductivities  at  different  dilutions  of  the  same  number  of 
molecules  placed  between  plates  which  are  the  same  distance  apart ; 
hence  the  name  molecular  conductivity.^ 

Now  the  molecular  conductivities  of  nearly  all  electrolytes  are 
found  to  increase  with  increasing  dilution  and  to  approach  definite 

*  The  lowering  of  the  freezing  point  produced  by  .01  gram-molecule  to  one 
liter  of  water  (that  is,  a  1/100  normal  solution)  is  .01857°  C.  for  sugar.  (See 
Whetham,  "  Theory  of  Solution,"  p.  320.)  Twice  this  value  is  .03714  and  three 
times  is  .05571.  The  observed  depressions  produced  by  some  of  the  substances 
which  could  split  up  into  only  two  ions  are  as  follows  :  KOH,  .0371  ;  HC1,  .0361 ; 
KC1,  .0360  ;  NaCl,  .0367.  The  depressions  produced  by  substances  which  could 
split  up  into  three  ions  are  illustrated  by  the  following  : 

H2S04(H,  H,  S04).0449,    Na2S04(Na,  Na,  SO4).0509, 
CaCl2(Ca,  Cl,  Cl).0504,   MgCl2(Mg,  Cl,  Cl).0508. 

t  This  illustration  is  taken  from  Walker,  "  Introduction  to  Physical  Chem- 
istry," p.  227. 


ELECTROLYTIC   CONDUCTION 


181 


Dilution 

Molecular 
Conductivity 

1 

69.5 

2 

79.7 

10 

86.5 

20 

89.7 

100 

96.2 

500 

99.8 

1,000 

100.8 

5,000 

101.8 

10,000 

102.9 

50,000 

102.8 

100,000 

102.4 

values  as  the  dilution  becomes  very  great.    This  is  evident  from 
the  following  table  of  observations  on  Nad  made  by  Kohlrausch. 

From  Clausius's  view  as  to  the  cause 
of  dissociation  the  significance  of  the 
fact  that  the  molecular  conductivity 
approaches  a  limiting  maximum  value 
at  infinite  dilution -is  as  follows.  The 
chance  of  a  molecule  being  split  up  by 
an  unusually  violent  impact  against 
a  water  molecule  is  the  same  at  all 
dilutions,  but  the  chance  of  unlike 
ions  reuniting  into  molecules  becomes 
infinitely  small  at  infinite  dilution. 
Hence  at  infinite  dilution  the  dissoci- 
ation ought  to  be  complete,  as  the  ex- 
periments on  osmotic  pressure  given 

in  the  note  on  page  180  indicate  is  the  case  for  some  at  least  of 
these  substances,  even  when  the  dilution  is  no  greater  than  100. 
Hence  it  was  suggested  by  Arrhenius  hi  1887  that  the  per  cent 
of  dissociation  of  all  electrolytes  at  infinite  dilution  might  be 
considered  100,  and  the  dissociations  at  any  dilution  determined 
solely  from  molecular  conductivity  measurements.  The  manner 
in  which  this  can  be  done  may  be  seen  from  the  following 
considerations  due  to  Arrhenius  and  Kohlrausch. 

Molecular  conductivity  can  vary  with  dilution  only  because  of 
the  variation  of  one  of  two  factors.  (1)  The  per  cent  of  the  mole- 
cules introduced  which  are  dissociated  into  ions  may  change  with 
the  dilution ;  (2)  the  speed  with  which  the  ions  move  through 
the  solution  may  change  with  dilution.  Now  if  a  change  in  dilu- 
tion caused  no  change  at  all  in  the  resistance  which  the  individual 
ions  experience  in  moving  through  the  solution,  that  is,  in  the 
ionic  speeds,  then  the  molecular  conductivities  at  two  dilutions 
would  obviously  be  proportional  to  the  number  of  ions  present  in 
each  case,  that  is,  to  the  percentage  of  the  total  number  of  mole- 
cules introduced  which  have  become  dissociated  into  ions.  If  m 
represents  the  molecular  conductivity  of  a  substance  at  any  given 
dilution,  and  mm  the  molecular  conductivity  at  infinite  dilution, 


182  ELECTEICITY,  SOUND,  AND  LIGHT 

and  if  n  and  n^  correspond  to  the  respective  number  of  ions  present 
in  the  two  cases,  then,  if  the  above  supposition  were  correct,  we 
should  have 

—  =  ^L.  n\ 

m<x>       n«>  '   ' 

On  the  other  hand,  if  the  dissociation  were  uninfluenced  by 
the  dilution,  but  if  the  resistance  which  the  ions  meet  in  moving 
through  the  solution  were  proportional  to  the  viscosity  of  the  solu- 
tion, then  obviously  the  ionic  speeds  would  be  inversely  propor- 
tional to  the  viscosities  at  the  dilutions  considered.  That  is,  if  77 
and  77^  represent  the  viscosity  coefficients  at  the  given  dilution 
and  at  infinite  dilution  respectively,  then  we  should  have 


moo        77 

If  both  the  dissociation  and  the  viscosity  change  with  dilution, 
we  have,  by  combining  the  two  variations  expressed  in  equations 

(1)  and  (2), 

mL  =  ^L^_  „ 

moo       n<x>    77  ' 

n        m     TJ    * 

or  -  =  ---  -  •  (4) 

Woo       moo  7700 

If  at  infinite  dilution  the  dissociation  is  complete,  then  obviously 
n/n<»  represents  the  fraction  of  all  the  molecules  introduced  which 
are  split  into  ions  at  the  dilution  corresponding  to  m.  This  is 
usually  represented  by  the  letter  a.  We  have,  then, 

«  =  -5L-5-.t  (5) 

m<»  7700 

As  a  matter  of  fact  the  dissociation  of  electrolytes  as  computed 
from  equation  (4)  does  not  agree  perfectly  with  the  dissociation 

*  The  molecular  conductivities  m  and  m^,  as  well  as  the  viscosity  coefficients 
17  and  77^,  all  refer  to  the  same  temperature. 

t  Arrhenius  omitted  the  term  V^,  and  wrote  merely  a  =  m/m^  for  any 
given  temperature.  Indeed,  since  for  dilutions  greater  than  8  the  ratio  y/ri^  is 
found  to  differ  but  little  from  unity,  it  is  not  improper  to  write  the  dissoci- 
ation equation  in  this  form,  provided  its  restriction  to  dilutions  larger  than 
about  8  is  understood. 


ELECTKOLYTIC   CONDUCTION 


183 


estimated  from  freezing-point  determinations,  although  in  general 
there  is  a  fair  agreement,  and  in  some  cases  an  exceedingly  close 
one.  In  view  of  these  uncertainties  it  must  be  said  that  the  dis- 
sociation theory  as  here  developed  is  not  yet  fully  established. 

120.  Methods  of  measuring  molecular  conductivity.  In  prac- 
tice molecular  conductivities  cannot  be  measured  in  the  simple 
manner  described  in  section  119,  in  which  the  immersed  area  of 
the  plates  increased  proportionally  to  the  dilution,  and  hence  the 
conductivity  changed  only  because  of  a  change  in  the  molecular 
conditions  due  to  the  dilution.  The  same  result  is  obtained  by 
measuring  the  resistance  of  the  solution  between 
plates  1000  sq.  cm.  in  area  and  1  cm.  apart,  and 
multiplying  the  reciprocal  of  this  resistance  by 
the  dilution.  If  the  molecular  conductivity  is  to 
be  obtained  in  absolute  units,  plates  of  1  sq.  cm. 
area  are  taken  1  cm.  apart ;  that  is,  the  absolute 
values  of  molecular  conductivity  are  -joVo  of 
those  obtained  as  described  above.  Since  the 
resistance  of  a  centimeter  cube  is  called  specific 
resistance  (see  p.  71)  and  since  specific  conduc- 
tivity is  the  reciprocal  of  specific  resistance,  it 
will  be  seen  that  molecular  conductivity  in  abso- 
lute units  is  the  dilution  divided  ly  the  specific 
resistance  of  the  solution  under  consideration. 

The  vessel  most  commonly  used  for  molecular- 
conductivity  experiments  is  shown  in  Figure  132. 
It  consists  of  two  platinum  electrodes  supported, 
as  shown,  in  a  vessel  made  of  a  kind  of  glass  which  is  especially 
insoluble  in  water.  If  only  comparisons  of  molecular  conduc- 
tivities are  desired,  they  are  made  by  introducing  the  different 
solutions  to  be  compared  succe'ssively  into  the  vessel  and  divid- 
ing the  given  dilutions  by  the  observed  resistances.  On  account, 
however,  of  the  fact  that  in  this  vessel  the  current  is  conducted 
to  some  extent  by  the  portion  of  the  solution  which  is  outside 
the  edges  of  the  plates,  an  absolute  determination  is  usually  made 
on  a  standard  solution  in  a  vessel  of  another  type,  and  then  com- 
parisons are  made  in  the  vessel  showTi  above.  However,  absolute 


FIG.  132 


184 


ELECTRICITY,  SOUND,  AND   LIGHT 


specific  resistance  determinations  of  approximate  correctness  may 
be  made  with  the  cell  of  Figure  132  by  multiplying  the  observed 
resistance  by  the  area  of  the  plates  and  dividing  by  the  distance 
between  them. 

The  measurement  of  the  resistance  is  made  by  making  the  vessel 
one  branch  of  a  Wheatstone's  bridge,  as  in  Figure  133.  Although 
the  electrodes  are  of  a  different  substance  from  that  of  the  metallic 
ion  of  the  electrolyte,  difficulties  due  to  the  back  E.M.F.  of  polari- 
zation are  eliminated  by  using  a  source  of  alternating  current  in 
place  of  the  battery  of  the  ordinary  bridge  scheme.  This  obviously 


FIG.  133 

allows  no  permanent  separation  of  the  elements  of  the  electrolyte 
at  the  electrodes.  The  source  of  alternating  current  is  the  second- 
ary of  a  small  induction  coil  C.  A  telephone  receiver  replaces  the 
galvanometer  of  the  ordinary  bridge  arrangement.  The  connec- 
tions of  Figure  133  differ  from  those  of  Figure  45,  page  63,  in  the 
fact  that  the  position  of  the  generator  and  of  the  detector  are  inter- 
changed. But,  as  we  have  seen,  this  does  not  alter  the  equation 
of  balance.* 

It  is  sometimes  difficult  in  an  arrangement  of  this  sort  to  get 
a  sharp  minimum.  This  is  probably  because  a  thin  film  of 
some  insulating  material,  presumably  gas,  collects  on  the  platinum 

*  That  this  is  so  is  evident  from  the  discussion  in  section  46.  The  reason  for 
this  interchange  is  given  in  the  footnote  on  page  71.  In  this  case  the  gener- 
ator resistance  is  that  of  the  secondary  of  the  small  induction  coil  and  is  higher 
than  the  resistance  of  an  ordinary  telephone  receiver. 


ELECTROLYTIC   CONDUCTION  185 

electrodes,  and  makes  each  of  them  act  as  a  condenser,  having  the 
gas  film  as  a  dielectric,  the  platinum  and  the  solution  as  the  two 
opposite  condenser  plates.  Such  condensers  in  series  with  the 
electrolytic  resistance  would  have  no  effect  upon  the  current,  pro- 
vided their  capacities  were  so  large  that  the  opposite  sides  of  the 
condensers  did  not  charge  to  any  appreciable  P.D.  before  the  applied 
P.D.  of  the  generator  reversed  its  direction.  If  this  were  not  the 
case,  the  introduction  of  such  condensers  would  cut  down  the 
alternating  current  flowing  through  the  branch  containing  them 
and  hence  destroy  the  balance.  Hence  these  difficulties  may  be 
eliminated  by  using  platinum  plates  of  very  great  surface,  for  then 
the  capacities  are  very  large.  The  method  usually  employed  for 
increasing  the  size  of  the  plates  is  to  cover  the  surfaces  of  the 
electrodes  with  spongy  deposits  of  platinum  called  "platinum 
black."  * 

EXPERIMENT  16 

Object.  To  compare  the  molecular  conductivities  of  some  salt,  for  example 
sodium  chloride  (XaCl),  at  different  dilutions. 

Directions.  Set  up  a  cell,  similar  to  that  of  Figure  132,  in  a  large  bath  of 
water  at  the  room  temperature.  Connect  as  in  the  diagram  of  Figure  133, 
using  a  meter  bridge  of  which  the  wire  is  Xo.  36  German  silver.  Prepare 
a  normal  solution  of  XaCl  either  by  weight  or  by  hydrometric  measurements 
with  a  Mohr's  balance, f  using  the  values  of  the  density  given  in  table  5 
of  the  Appendix. 

Introduce  two  pipettes  full  of  this  solution  into  the  cell.  Measure  the 
resistance  and  take  its  reciprocal.  Carefully  remove  one  pipette  full  of 
solution  and  replace  by  a  pipette  of  distilled  water  at  room  temperature. 
The  dilution  is  then  2.  Stir  by  moving  the  electrodes  up  and  down,  but 
be  very  careful  not  to  alter  in  any  way  their  distance  apart.  Then  measure 
the  resistance  again.  Similarly  introduce  another  pipette  full  of  water, 
and  measure  the  resistance  at  the  dilution  4.  Continue  in  this  way  until 
a  dilution  of  2048  is  reached. 

*  To  platinize  the  electrodes  place  them  in  a  3  per  cent  solution,  by  weight, 
of  platinum  chloride  to  which  is  added  a  drop  or  two  of  lead  acetate  solution. 
Pass  a  current  through  this  cell  of  such  a  value  that  the  bubbles  rise  freely 
from  the  solution,  then  reverse  its  direction  occasionally.  Wash  in  distilled 
water  and  allow  the  electrodes  to  soak  in  the  water  for  some  time  before 
using  them. 

t  See  "Mechanics,  Molecular  Physics,  and  Heat,"  p.  175. 


186 


ELECTRICITY,  SOUND,  AND  LIGHT 


Empty  and  rinse  the  cell  and  determine  the  resistance  of  the  distilled 
water  used.  Take  its  reciprocal  and  subtract  from  the  reciprocals  of  the 
resistances  obtained  above,  then  multiply  by  the  corresponding  dilutions 
to  obtain  quantities  proportional  to  the  molecular  conductivities. 

Assume  the  number  thus  corrected  which  corresponds  to  a  dilution  of 
2048  as  »j  ,  and  find  a  for  the  other  dilutions  used  from  equation  (5) 
on  page  182.  The  values  of  y  and  TJX  may  be  taken  from  table  6  of  the 
Appendix,  and  the  ratio  assumed  correct  for  the  temperature  at  which 
the  experiment  was  performed. 


EXAMPLE 

The  distance  between  the  plates  of  the  cell  used  was  about  1  cm.,  and 
the  diameter  of  the  plates  about  2.5  cm.  The  resistance  of  the  distilled 
water  used  was  found  to  be  11,143  ohms.  Hence  its  conductivity  was  pro- 
portional to  .0000897.  This  correction  was  neglected  for  dilutions  less  than 
64.  The  salt  used  was  ZnSO4.  The  values  observed  and  calculated  follow. 


Dilution 

Resistance 

Corrected 
Conductivity 

1? 

'VoO 

«  or  the  % 
Dissociation 

1 

2.27  ohms 

.480 

1.362 

.844 

2 

3.88 

.515 

1.166 

.848 

4 

7.16 

.559 

1.080 

.852 

8 

13.64 

.586 

1.040 

.859 

16 

26.17 

.611 

1.020 

.879 

32 

50.41 

.635 

1.010 

.904 

64 

97.57 

.655 

1.005 

.928 

128 

189.6 

.671 

1. 

.946 

256 

368.1 

.673 

1. 

.949 

512 

706.7 

.679 

1. 

.958 

1024 

1293. 

.701 

1. 

.989 

2048 

2294. 

.709 

1. 

4096 

3747. 

.709 

1. 

oc  .709 


temperature  22C 


CHAPTEE  XVII 
VELOCITY  OF  SOUND  IN  AIR 

121.  Sound.    Sound  is  the  name  applied  to  the  sensations  of  the 
auditory  nerves.    It  is  the  result  of  the  propagation  through  the 
air  to  the  ear  of  a  disturbance  produced  by  the  sounding  body. 
Thus  when  a  pistol  is  fired  the  layer  of  air  surrounding  it  is  sud- 
denly pushed  outward  and  compressed  by  the  expanding  gases 
which  emerge  from  the  pistol.    This  compression  is  transmitted 
from  layer  to  layer  and  constitutes  a  compressional  pulse  traveling 
outward  from  the  pistol.    It  is  this  pulse  which,  giving  up  a  por- 
tion of  its  energy  to  the  eardrum,  results  in  a  stimulation  of  the 
nerves  of  the  ear  and  the  sensation  of  sound.    For  the  purposes  of 
physics  the  subject  of  Sound  is   commonly  limited  to  the  study 
of  the  causes  and  nature  of  these  pulses  and  the  mechanism  of 
their  transmission. 

122.  Velocity  of  a  compressional  pulse  in  any  elastic  medium. 
Consider,   then,  the  velocity   of  propagation   of   a  compressional 
pulse  in  an  elastic  medium.    For  convenience  the  medium  will  be 
assumed  to  be  contained  in  a  tube  of  one  square  centimeter  cross 
section  and  of  infinite  length.    Let  the  density  of  the  medium  be 


\\\\\\\\\ 

O                                 )   1 

'\Ll 

2 

3 

4 

5 

6 

7 

FIG.  134 

represented  by  p  and  the  pressure  under  which  it  stands  before 
any  compression  is  produced  by  p.  For  convenience  of  analysis 
let  the  medium  be  conceived  to  be  divided  into  centimeter  cubes, 
1,  2,  3,  4,  5,  etc.  (Fig.  134).  Let  one  end  of  the  tube  be  closed  by 
a  frictionless,  weightless  piston.  Now  let  the  piston  be  suddenly 

187 


188  ELECTRICITY,  SOUND,  AND  LIGHT 

started  forward  by  the  application  to  it  of  a  constant  pressure 
slightly  greater  than  p,  viz.  p  +  dp.  There  will  thus  be  started 
down  the  tube  a  compressional  pulse  which  will  travel  through 
the  medium  with  a  velocity  which  will  be  represented  by  S.  It 
is  this  velocity  which  it  is  desired  to  determine. 

As  soon  as  the  piston  begins  to  move  forward,  cube  1  begins  to 
be  compressed.  The  pressure  inside  of  this  cube  rises.  When  this 
pressure  has  reached  the  value  p  +  dp,  the  cube  will  cease  to  be 
compressed  and  will  thenceforth  merely  transmit  pressure  to  cube  2. 
Call  dv  the  amount  of  compression,  measured  in  fractions  of  a  cen- 
timeter, which  cube  1  thus  experiences.  Under  the  action  of  the 
pressure  p  +  dp,  which  is  thus  transmitted  by  cube  1  to  cube  2,  the 
latter  will  also  be  compressed  an  amount  dv  and  will  thereafter 
merely  transmit  the  pressure  p  +  dp  to  cube  3.  Similar  reasoning 
may  be  applied  to  the  remaining  cubes  in  their  numerical  order. 
Since  the  cross  section  of  the  tube  is  1  sq.  cm.,  while  any  particular 
cube,  such  as  6,  is  being  compressed  the  amount  dv  cc.,  the  piston 
will  move  forward  dv  cm.,  and  each  of  the  cubes  previously  com- 
pressed, namely  1,  2,  3,  4,  and  5,  will  move  forward  dv  cm.  .  Thus, 
as  the  pulse  moves  down  the  tube,  the  piston,  and  with  it  all 
the  compressed  cubes,  will  move  uniformly  forward.  Since  S  repre- 
sents the  velocity  of  the  pulse,  and  since  the  cubes  were  chosen 
one  centimeter  long,  it  follows  that  in  one  second  each  of  S  cubes 
wTill  experience  the  compression  dv.  During  that  second  the 
piston  will  therefore  have  moved  forward  Sdv  cm.  That  is,  the 
velocity  of  the  piston  is  Sdv  cm.  per  second.  Evidently  this  is 
also  the  expression  for  the  velocity  with  which  all  of  the  cubes 
which  have  been  compressed  are  moving  forward  at  the  end  of 
a  second. 

Now  the  velocity  with  which  the  pulse  moves  forward  may  be 
found  by  an  application  of  Newton's  principle  of  work  *  as  follows. 
In  the  above  operation  the  acting  force  is  the  pressure  applied  to 
the  piston,  viz.  p  +  dp.  The  work  done  by  this  force  in  one  second 
is  (p  +  dp)  Sdv.  The  effects  of  the  action  of  the  force  are  of  two 
kinds :  (1)  the  compression  of  all  the  gas  contained  in  S  cubes 

*  This  method  of  analysis  is  found  in  Edser's  "  Heat  for  Advanced  Students." 


VELOCITY  OF  SOUND   IN  AIR  189 

from  a  condition  in  which  it  exerts  a  pressure  p  to  one  in  which 
it  exerts  a  pressure  p  +  dp  ;  and  (2)  the  communication  of  a  veloc- 
ity Sdv  to  all  the  mass  contained  in  S  cubes.  The  mean  force 
overcome  in  the  first  operation  is  half  the  sum  of  the  initial  and 
final  pressure,  namely  p  +  dp/2,  and  the  work  done  against  this 
force  is  (p  +  dp/  2)  Sdv.  In  the  second  operation,  since  the  mass 
of  each  cube  is  p,  and  since  in  one  second  S  cubes  are  set  in  motion 
with  a  velocity  Sdv,  the  kinetic  energy  imparted  per  second  is 
\-  pS  (Sdv)2.  Now  by  the  scholium  to  Newton's  third  law  of 
motion,  the  work  done  by  the  acting  force  is  equal  to  the  work 
done  against  the  resisting  forces  ;  i.e.  it  is  equal  to  the  sum  of  the 
potential  and  kinetic  energies  imparted  to  the  gas.  Hence  we  have 


(p  +  dp)  Sdv  =   p  +         Sdv  +  £  •  pS(Sdv)2.  (1) 

dp  _  pS2dv 


.'.  S'2  =  -t-,     or        S  =  —  •  (2) 

pdv  \J  p 

Now  since  we  are  dealing  with  unit  cubes  the  quantity  dp/dv 
is  the  force  applied  per  unit  area  divided  by  the  change  in  volume 
per  unit  volume,  and  this  is  by  definition  the  volume  modulus  of 
elasticity  E  of  the  medium.  Hence,  finally,  we  have  for  the  velocity 
of  a  compressional  pulse  in  any  medium, 


Since  this  expression  involves  only  the  constants  E  and  p  of  the 
medium,  it  is  evident  that  a  pulse  once  started  will  travel  on  and 
on  down  the  tube  at  a  rate  which  has  nothing  whatever  to  do 
either  with  the  size  or  shape  of  the  tube,  or  with  whether  the  pis- 
ton continues  to  move  forward  or  not.  In  other  words,  we  have 
deduced  a  general  expression  for  the  velocity  of  a  compressional 
pulse  in  any  medium.  The  expression  must  hold  for  the  velocity 
of  propagation  of  a  sound  pulse  which  originates  at  a  point  within 
any  medium  and  spreads  radially  from  the  center  of  disturbance  ; 


190  ELECTRICITY,  SOUND,  AND  LIGHT 

for,  in  this  case,  as  in  the  case  just  discussed,  the  pulse  is  one 
of  pure  compression,  since  the  particles  are  free  to  move  only 
in  one  direction,  namely  along  radii  emanating  from  the  point  of 
disturbance. 

It  will  be  interesting  to  see  how  well  the  results  obtained  by 
the  use  of  this  formula,  which  has  been  deduced  from  purely  theo- 
retical considerations,  agree  with  the  results  of  direct  experiment. 
In  1893  Aiuagat  in  Paris  made  some  experiments  on  the  compres- 
sibility of  water,  and  found  that  at  10°C.  a  change  in  pressure  from 
1  atmosphere  to  50  atmospheres  produced  a  change  in  volume  from 
1  cc.  to  .99757  cc.*  From  this  data  we  get  as  the  theoretical  value 
of  the  velocity  of  sound  in  water,  all  quantities  being  expressed  in 
absolute  units, 


S 


=     149x76x13.6x981  =1 
N         lx.00243 


The  result  of  direct  measurement  made  at  8°C.  in  Lake  Geneva 
in  1827  by  Colladon  and  Sturm  gave  1435  meters  per  second.  J 
The  difference  between  the  two  values  is  well  within  the  limits 
of  observational  error. 

123.  A  train  of  waves.  If  in  the  case  of  the  tube  and  piston 
of  Figure  134  the  applied  pressure  had  been  p  —  dp  instead  of 
p  +  dp,  the  piston  would  have  started  back  instead  of  forward,  and 
cube  1  would  then  have  expanded  until  its  pressure  reached  the 
value  p  —  dp,  which  expansion  would  have  been  followed  by  a 
similar  expansion  of  cube  2,  etc.  Thus  a  pulse  of  rarefaction  in- 
stead of  one  of  condensation  would  have  traveled  down  the  tube, 
and  reasoning  in  every  respect  identical  with  the  above  shows 
that  the  velocity  of  this  pulse  also  would  have  been  ^/E/p. 

It  is  important  to  observe  that  in  a  pulse  of  condensation  tlie  par- 
ticles are  always  moving  in  the  same  direction  as  the  pulse ;  whereas 
in  a  pulse  of  rarefaction  the  direction  of  motion  of  the  particles  is 
always  opposite  to  the  direction  of  propagation  of  the  pulse. 

*  Wiillner,  "Experimental  Physik,"  Vol.  II,  ed.  5,  p.  100. 
t  981  is  the  value  rdf  the,  acceleration  of  gravity  in  Paris. 
\  Wiillner,  "  Experimental  Physik,"  Vol.  I,  ed.  5,  p.  955. 


VELOCITY  OF  SOUND  IN  AIR 


191 


If  the  piston  is  moved  alternately  forward  and  back  at  regular 
intervals,  a  succession  of  compressions  and  rarefactions  will  follow 
one  another  down  the  tube.  Such  a  succession  of  compressions 
and  rarefactions  is  called  a  train  of  waves.  In  this  case  it  is  evi- 
dent that  the  motions  of  all  of  the  particles  of  the  medium  follow, 
in  succession,  exactly  the  motions  of  the  piston ;  that  is,  each  par- 
ticle moves  forward  during  just  the  interval  of  time  during  which 
the  piston  is  moving  forward,  and  back  during  just  the  interval 
during  which  the  piston  is  moving  back.  If  then  the  piston  is 
replaced  by  the  vibrating  prong  of  a  tuning  fork  or  by  any  body 
which  vibrates  under  the  influences  of  its  own  elasticity,  the  back- 
ward motion  will  begin  at  exactly  the  instant  at  which  the  forward 
motion  ends,  and  hence  at  the  end  of  one  complete  vibration  of 

ABC  c  a 


FIG.  135 


the  prong,  i.e.  at  the  end  of  the  time  required  for  the  prong 
to  go  from  A  to  C  and  back  again  to  A  (Fig.  135),  the  whole 
of  the  medium  between  B  and  some  point  a  to  which  a  sound 
pulse  travels  during  the  period  of  one  vibration,  may  be  divided 
into  two  equal  parts,  ac  and  cB,  such  that  all  the  layers  between  c 
and  a  are  moving  forward  and  are  in  a  state  of  compression,  while 
all  the  layers  between  c  and  B  are  moving  back  and  are  in  a  state 
of  rarefaction.  The  relative  velocities  with  which  the  layers  are 
moving  forward  or  back  at  the  various  points  between  B  and  a 
are  represented  by  the  lengths  of  the  arrows  in  the  figure.  As  the 
fork  continues  to  vibrate  the  whole  region  about  it  becomes  filled 
with  a  series  of  such  waves,  each  consisting,  as  above,  of  a  conden- 
sation and  a  rarefaction  (see  Fig.  135).  The  distance  between  the 
beginnings  of  two  successive  condensations  or  two  successive  rare- 
factions, or  in  general  the  distance  bet\^Jen  any  two  particles  which 
are  in  the  same  condition  or  phase  of  vibration,  is  called  a  wave 


192 


ELECTRICITY,  SOUND,  AND  LIGHT 


length.  It  is  obvious  that  if  S  represent  the  velocity  of  sound, 
n  the  number  of  vibrations  of  the  fork  per  second,  and  X  the 
wave  length,  then  the  following  relation  holds : 


S  =  n\. 


(3) 


The  only  distinction  between  a  musical  note  and  a  mere  noise 
is  that  the  former  consists  of  a  train  of  waves,  while  the  latter  con- 
sists either  of  single  pulses,  or  of  irregularly  timed  pulses. 

124.  Newton's  deduction  of  the  velocity  of  sound  in  air.    If  dp 

represent  the  change  in  pressure  which  is  applied  to  any  body  of 

volume  F,  and  if  dV  be  the  change  in 

I  p>  volume  produced  by  dp,  then,  by  defini- 
tion, the  volume  modulus  of  elasticity  E 
is  given  by 

E=LTT/  (4) 


1 


T 


1 


FIG.  136 


Let  the  body  be  a  gas  of  volume  V  and 
pressure  p,  and   let    the   applied  force 
change  the  volume  of  the  gas  to  V'  and  its  pressure  to  p'  (see 

Fig.  136).    Then  , 

dp  _pf-p 

dV      V-V 


But  if  the  heat  which  is  developed  by  the  compression  is  allowed 
to  pass  off,  so  that  the  final  temperature  is  the  same  as  the  initial 
(i.e.  if  the  change  is  isothermal),  then  Boyle's  law  may  be  applied. 
This  gives  p/p'  =  V /  V,  or  V  =  p  V/p'.  Substitution  of  V  in  (5) 

gives 

r  ._  ^  (6) 


Now  if,  as  is  the  case  with  sound  waves,  pf  differs  from  p  by  an 
immeasurably  small  quantity,*  then  it  is  possible  to  replace  p'  byj?. 


*  Lord  Rayleigh  estimates  that  for  certain  feeble,  though  distinctly  audible, 
sounds  dp  is  not  more  than  6  x  10~9  atmospheres. 


VELOCITY  OF  SOUND  IN  AIE  193 

In  other  words,  the  volume  coefficient  of  elasticity  of  a  gas  for 
isothermal  changes  is  the  pressure  under  which  the  gas  stands. 
Hence  if  a  sound  wave  produces  no  temperature  changes  in  a  gas 
we  have 


velocity  =  .  (7) 

N  density 

If  the  gas  is  air  under  ordinary  conditions,  then  p  is  simply  the 
barometric  pressure,  which  must,  of  course,  be  reduced  to  dynes  if 
S  is  to  be  determined  in  centimeters.  This  theoretical  value  for 
the  velocity  of  sound  in  air,  first  deduced  by  Newton  (1687),  does 
not  agree  with  experimental  determinations,  for  the  simple  reason 
that  the  compression  produced  by  a  sound  wave  does  not  take 
place  isothermally,  as  Newton  assumed  it  to  do. 

125.  Correct  formula  for  velocity  of  sound  in  air.    It  was  a 

hundred  years  after  Newton's  time  when  La  Place  first  pointed 
out  that  a  gas  is  heated  by  a  compressional  wave,  and  on  account 
of  the  great  velocity  of  sound  the  heat  thus  developed  has  no  time 
to  diffuse  before  the  wave  is  past.  When  all  the  heat  developed 
by  compression  is  retained  in  the  compressed  body  the  change  in 
pressure  is  said  to  take  place  adiabatically.  Since  bodies  expand 
with  heat,  it  is  evident  that  the  change  in  volume  produced  in  a 
gas  by  a  given  change  in  pressure  will  be  greater  if  the  heat  of 
compression  is  allowed  to  pass  off  than  if  it  is  retained.  That  is, 
d  V/  V  is  less  if  the  change  is  adiabatic  than  if  it  is  isothermal. 
Hence  the  quantity  E  =  (dp/d  V]  V  is  greater  for  an  adiabatic 
change.  Analysis  which  is  beyond  the  scope  of  this  text  shows 
that  the  bulk  modulus  of  a  gas  for  an  adiabatic  change  is  equal  to 
its  bulk  modulus  for  an  isothermal  change  multiplied  by  a  factor  7. 
This  factor  7  is  the  ratio  of  the  specific  heat  of  the  gas  at  con- 
stant pressure  and  its  specific  heat  at  constant  volume.  It  is 
always  greater  than  unity  ;  for  air  it  has  a  value  1.403  ;  for  carbon 
dioxide  (CO2),  a  value  1.30.  Hence  the  theoretical  value  for  the 
velocity  of  sound  in  a  gas  is 


194  ELECTRICITY,  SOUND,  AND  LIGHT 

126.  Effect  of  temperature  upon  the  velocity  of  sound.    If  the 

pressure  is  changed  without  a  change  in  temperature,  then  by 
Boyle's  law  the  density  change  is  directly  proportional  to  the 
pressure  change ;  that  is,  p/p  =  constant.  Hence  the  velocity  of 
sound  in  gases  is  wholly  independent  of  pressure. 

The  dependence  of  the  velocity  upon  the  temperature  of  the  gas 
may  be  found  by  the  following  reasoning.  Suppose  the  pressure 
of  the  gas  to  remain  constant.  Let  St  represent  the  velocity  at  a 
temperature  of  t°  C.,  and  pt  the  corresponding  density.  Similarly, 
let  S0  and  p0  represent  the  velocity  and  density  at  0°.  The  veloc- 
ity at  the  temperature  t°  is  given  by  the  equation  St  =  ^yp/pt,  the 
velocity  at  0°  by  the  equation  S0=^/yp/p0.  Therefore 

(9) 

Now  by  Gay-Lussac's  law  the  density  of  a  given  gas  is  inversely 
proportional  to  its  absolute  temperature.  Hence 

St 


in  which  T0  is  273  and  T  is  the  temperature  of  the  gas  on 
the  Centigrade  scale  plus  273.  By  substituting  in  this  formula 
274  for  T  and  for  S0  the  best  determination  of  the  velocity  of 
sound  at  0°,  namely  33,127  cm.,  it  will  be  seen  that  the  velocity 
of  sound  increases  about  60  cm.  for  each  degree  Centigrade  of  rise 
in  temperature. 

127.  Reflection  of  wave  trains  produced  by  a  change  in  the 
density  of  the  medium.  Eeturning  to  the  consideration  of  the 
tube  and  piston,  suppose  the  latter  to  move  forward  suddenly  a 
short  distance  and  then  stop.  A  compressional  pulse  will  start 
down  the  tube.  The  progression  of  this  pulse  is  due  to  the  com- 
munication from  layer  to  layer  of  the  motion  which  is  imparted 
by  the  piston  to  the  layer  next  it.  The  case  is  precisely  analogous 
to  that  of  the  collision  of  perfectly  elastic  equal  balls.  So  long  as 


VELOCITY  OF  SOUND  IN  AIR 


195 


FIG.  137 


the  pulse  travels  in  a  medium  of  uniform  density  each  layer  of 
particles  gives  up  all  its  motion  to  the  next  layer,  which  is  pre- 
cisely like  it,  just  as  a  moving  elastic  ball  striking  a  stationary 
elastic  ball  of  the  same  size  gives  up  all  of  its  motion  to  the 
stationary  ball  and  itself  comes  to  rest.  But  if  the  pulse  at  some 
point  A  (Fig.  137)  strikes  a  denser  medium  D,  the  case  becomes 
analogous  to  the  impact  of  p  A 

a  lighter  ball  upon  a  heav- 
ier one.  The  lighter  layer  CZZ 
adjoining  A  on  the  left,  in- 
stead of  coming  to  rest  in 
the  impact,  reverses  its  mo- 
tion and  starts  back.  This  backward  motion  is  communicated 
from  layer  to  layer  so  that  a  reflected  compressional  pulse  travels 
from  A  back  toward  P.  Let  the  long  arrow  represent  the  direction 
of  the  pulse  and  the  short  one  the  direction  of  motion  of  the  par- 
ticles in  the  pulse.  It  is  evident  that  the  direction  of  motion  of 
the  particles  with  respect  to  the  direction  of  motion  of  the  pulse 
is  the  same  in  the  returning  as  in  the  advancing  pulse.  In  other 
words,  a  pulse  of  condensation  is  reflected  from  a  denser  medium 
as  a  pulse  of  condensation. 

But  suppose  the  medium  to  the  right  of  A  is  less  dense  than 

that  in  the  tube  (see  R,  Fig.  138),  then  the  case  is  analogous  to 

the  impact  of  a   heavy   ball  upon  a  light  one.    Until  the  pulse 

reached  A  each  layer  in  the  tube  gave  up  its  motion  to  the  next 

P  A      and  itself  came  to  rest ;  but 

the  layer  at  A,  instead  of 
coming  to  rest  after  the  im- 
pact, continues  to  move  for- 
ward and  thus  produces  a 
rarefaction,  i.e.  a  diminution 
of  pressure  at  A.  The  excess  of  pressure  in  the  layer  to  the  left 
of  A  then  drives  particles  toward  the  right.  Thus  a  pulse  of  rare- 
faction moves  back  from  A  toward  P.  In  other  words,  a  pulse 
of  condensation  is  reflected  from  a  rarer  medium  as  a  pulse  of 
rarefaction.  In  the  pulse  approaching  A  the  particles  move  in 
the  direction  of  propagation  of  the  wave.  In  the  pulse  receding 


FIG.  138 


196  ELECTEICITY,  SOUND,  AND  LIGHT 

from  A  the  particles  move  in  a  direction  opposite  to  that  of 
propagation  of  the  wave  (see  arrows,  Fig.  138).  Since  the  pulse 
changes  instantly  in  the  reflection  from  a  condensation  to  a  rare- 
faction, and  since  in  a  train  of  waves  a  rarefaction  always  follows 
just  one  half  wave  length  behind  a  condensation,  a  wave  is  said  to 
experience  a  loss  of  one  half  wave  length  in  reflection  from  a  rarer 
medium. 

The  same  process  of  reasoning  shows  that  a  wave  of  rarefaction 
is  reflected  from  a  denser  medium  as  a  wave  of  rarefaction,  but 
from  a  rarer  medium  as  a  wave  of  condensation. 

128.  Resonance  of  vibrating  air  columns.  When  an  air  wave 
traveling  along  a  pipe  reaches  the  open  end  it  experiences  the  same 
sort  of  reflection  as  though  it  passed  from  a  denser  to  a  rarer 
medium.  This  statement  can  be  easily  proved  experimentally 

(see  sect.  130).     It 

2  2  1  i  °  is  also  evident  from 

the  theoretical  con- 
sideration'that  as 
soon    as    the  wave 
— >          < —  — >          < —  — 1>     B      reaches   a  point  at 

FIG.  139  •  which   lateral   ex- 

pansion is  possible, 

the  forward  movement  of  the  particles  is  greater  than  inside  the 
tube,  where  lateral  expansion  is  not  possible.  This  increased  for- 
ward movement  at  the  end  of  the  tube  means  a  wave  of  rarefac- 
tion starting  back  in  the  tube. 

Consider  now  a  train  of  waves  approaching  the  open  end  of  a 
pipe  the  other  end  of  which  is  closed  (Fig.  139).  In  the  conden- 
sations of  the  advancing  train  of  waves  the  motions  of  the  parti- 
cles are  all  in  the  direction  of  motion  of  the  wave,  i.e.  from  left 
to  right.  In  the  rarefactions  they  are  in  the  opposite  direction. 
Suppose  the  pipe  to  have  a  length  of  exactly  one  fourth  wave 
length.  Then  the  condensation  which  is  marked  0  will  move  down 
the  pipe  and  be  reflected  at  the  closed  end  as  a  condensation,  i.e. 
as  a  motion  of  the  particles  now  from  right  to  left.  It  will  obvi- 
ously return  to  the  open  end  at  the  exact  instant  at  which  the 
rarefaction  which  is  marked  J,  and  which  also  consists  of  a  motion 


VELOCITY  OF  SOUND  IN  AIR  197 

of  the  particles  from  right  to  left,  reaches  the  open  end  of  the  pipe. 
Since  the  reflected  condensation  which  is  returning  from  the  closed 
end  of  the  pipe  now  undergoes  reflection  at  the  open  end  as  a  rare- 
faction, i.e.  as  a  motion  of  the  particles  from  right  to  left,  the 
newly  reflected  wave  of  rarefaction  which  starts  back  down  the 
pipe  unites  with  the  rarefaction  marked  J-  which  is  just  entering 
the  pipe,  and  a  wave  of  rarefaction  of  increased  amplitude  is  the 
result.  This  wave  is  reflected  at  the  closed  end  as  a  rarefaction 
(motion  from  left  to  right),  and  again  at  the  open  end  as  a  con- 
densation (motion  from  left  to  right),  exactly  in  time  to  unite 
with  the  condensation  marked  1  as  it  enters  the  pipe.  Thus  by 
this  process  of  continuous  union  of  direct  and  reflected  waves  the 
motion  in  the  pipe  becomes  larger  and  larger  until  it  may  be  hun- 
dreds of  times  as  large  as  the  motion  of  the  particles  in  the  original 
wave.  In  fact,  there  would  be  no  limit  to  the  amplitude  of  the 
waves  traveling  up  and  down  the  pipe  if  at  each  end  the  motion 
were  not  partially  given  up  to  the  outside  air.  The  pipe  thus 
becomes  in  a  way  the  source  of  sound.  The  phenomenon  is  called 
resonance. 

If  the  pipe  had  been  only  a  trifle  longer  or  shorter  than  one 
fourth  wave  length,  the  error  in  the  coincidence  of  the  first  reflected 
wave  with  J-  would  have  been  but  slight.  Since,  however,  the 
reflected  waves  are  now  obliged  to  travel,  each  time  they  go  up 
and  down  the  pipe,  a  distance  a  little  too  great  or  too  small,  it 
takes  them  but  a  short  time  to  get  completely  out  of  step  with 
the  advancing  waves.  In  this  condition  the  direct  and  reflected 
waves  interfere  with  rather  than  assist  each  other,  and  no  reso- 
nance is  possible.  This  explains  why,  when  the  length  is  but  a 
trifle  more  or  less  than  the  right  amount,  very  little  resonance  is 
obtained. 

If,  however,  the  pipe  is  continually  lengthened,  other  resonant 
lengths  will  be  reached.  The  length  above  chosen  was  one  which 
permitted  0  to  return  to  the  mouth  of  the  pipe  exactly  in  time 
to  unite  with  \.  This  length  is  manifestly  the  shortest  possible 
resonant  length.  It  is  clear  that  the  next  possible  resonant  length 
is  one  which  permits, 0  to  return  exactly  in  time  to  unite  with  f. 
Since  |  is  one  wave  length  behind  \,  the  second  resonant  pipe 


198  ELECTRICITY,  SOUND,  AND  LIGHT 

length  must  be  one  half  wave  length  greater  than  the  first;  i.e 
it  must  be  three  fourths  wave  length.  Similarly,  it  is  possible 
to  obtain  resonance  when  the  pipe  length  is  five  fourths  wave 
length,  seven  fourths,  nine  fourths,  and  so  on. 

Experiment  shows  that  the  theoretical  values  of  the  pipe  length 
required  for  the  fundamental  resonance,  namely  one  fourth  wave 
length,  is  slightly  too  large.    The  discrep- 
ancy is  explained  as  follows.    If  the  reflec- 
\      tion  of  a  wave  traveling  down  the  open 
/      pipe  ac  (Fig.  140)  took  place  exactly  in 
the  plane  cd,  the  resonant  length  would 
FIG.  140  be  exactly  one  fourth  wave  length ;  but 

in  reality  the  point  of  free  lateral  expan- 
sion is  not  reached  at  the  instant  at  which  the  wave  reaches  cd ; 
hence  the  reflection  does  not  occur  until  the  wave  has  pushed 
out  a  short  distance  beyond  cd.  Thus  the  true  pipe  length  is 
slightly  greater  than  the  apparent  length.  The  amount  of  this 
correction  which  must  be  applied  at  the  open  end  is  estimated 
by  Eayleigh  as  the  radius  of  the  pipe,  but  in  glass  pipes  it 
is  somewhat  less  than  this. 


EXPERIMENT  17 

Object.  To  find  the  velocity  of  sound  in  air  from  resonance  experiments 
upon  a  closed  pipe,  and  to  find  the  correction  which  must  be  applied  to  the 
open  end  of  a  pipe. 

Directions.  The  pipe  to  be  used  consists  of  a  long  glass  tube  A  (Fig. 
141),  one  end  of  which  is  closed  by  water  admitted  by  a  tube  from  the 
vessel  B.  The  length  of  the  column  of  air  contained  in  this  pipe  may 
be  varied  by  changing  the  height  of  B.  A  rubber  band  around  A  may  be 
slipped  into  any  position  along  its  length.  A  train  of  waves  from  a  tuning 
fork  of  known  rate  is  caused  to  enter  the  pipe  by  holding  the  fork  over  the 
mouth  of  the  tube. 

Set  the  fork  T  in  vibration  by  striking  it  with  a  rubber  mallet,  and 
hold  it  over  the  mouth  of  the  tube.  It  is  not  important  in  this  part  of 
the  experiment  that  the  fork  be  held  farther  than  three  or  four  milli- 
meters from  the  end  of  the  tube,  provided  it  is  always  held  in  precisely 
the  same  position.  Taking  especial  care  about  this  point,  raise  or  lower 


VELOCITY  OF  SOUND  IX  AIR 


199 


the  vessel  B  and  thus  change  the  level  of  the  water  in  A  until  the  note 
of  the  fork  is  strongly  reenforced.  By  causing  the  water  to  rise  and  fall 
rapidly  several  times  in  the  vicinity  of  the  position  of  reenforcement,  the 
length  of  maximum  resonance  can  be  fairly  accurately  obtained.  Set  the 
rubber  band  at  the  level  of  the  water  in  the  tube  and  measure  to  it  from 

the  top  of  the  tube.  In  this 
way  determine  two  or  three 
successive  reenforcement 
points.  If  then  lv  12,  and  ls 
represent  the  first,  second, 
and  third  measured  resonant 
lengths,  x  the  unknown  cor- 
rection which  must  be  applied 
because  the  reflection  at  the 
open  end  does  not  take  place 
exactly  in  the  plane  no,  the 
preceding  theory  gives 

(1)  1 1  +  x  =  i  wave  length, 

(2)  12  +  x  =  |  wave  length, 
1                (3)  /3  +  x  -  f  wave  length. 

Eliminating  x  both  from  (1)  and  (2),  and  from 
(1)  and  (3),  gives 


n             o 

- 

A 

S 

- 

^=T=J 

1 

f3 

anc 

0) 

and 

=~_ 

1=7 

Frc 
the 
ma] 
tioi 
this 
bar 

FIG.  141 


From  the  mean  of  these  two  values  of  A.  and  from 
e  of  the  fork  n,  which  should  be  found 
marked  upon  it,  determine  with  the  aid  of  the  equa- 
tion S  =  n\  the  velocity  of  sound  in  air.  Compare 
this  with  the  theoretical  value,  deduced  from  the 
barometer  reading  and  the  density  of  air  for  the 
existing  temperature  and  pressure,  by  substitution 
in  the  equation 

c_    yp 
&  —  \  i —  • 

\  p 


Xext  repeat  very  carefully  the  determination  of  the  first  resonant  length, 
the  fork  being  now  held  at  least  as  far  away  from  the  end  as  the  radius  of 
the  tube.  By  subtracting  this  length  from  a  true  fourth  wave  length,  as 
determined  above,  find  the  correction  which  must  be  applied  at  the  open 
end  of  a  pipe  to  make  the  first  resonant  length  ^X.  This  might  have 
been  found  from  the  preceding  measurements  had  not  the  fork  been  held 
closer  to  the  end  than  the  radius  of  the  tube.  This  was  done  for  the  sake 


200  ELECTRICITY,  SOUND,  AND   LIGHT 

of  making  it  easier  to  locate  accurately  the  second  and  third  resonance 
points  which  otherwise  would  have  shown  but  feeble  reinforcement. 
Express  the  correction  x  as  a  fractional  portion  of  the  radius  of  the  pipe. 


EXAMPLE 

The  experiment  was  performed  at  a  temperature  of  24.0°  C.  and  a  cor- 
rected barometric  pressure  of  74.63  cm.  The  density  of  air  at  this  tem- 
perature and  pressure  was  found  from  Table  2  in  the  Appendix  to  be 
.001167.  The  pressure  p,  expressed  in  dynes  per  square  centimeter,  was 
found  by  multiplying  74.63  by  13.596,  the  density  of  mercury  at  0°,  and 

I~~J 
by  g.     Hence  S,  computed  from  the  theoretical  relation  S  = -i/y— ,was 

34,570  cm.  per  second. 

The  resonant  lengths  observed  were  15.9,  49.8,  and  84.1  cm.  Hence 
the  average  |  wave  length  was  34.0  cm.  The  fork  was  marked  n  =  500  ; 
but  when  it  was  sounded  simultaneously  with  a  standard  512  fork,  and 
the  two  held  close  to  the  ear,  it  was  found  that  there  were  four  beats  a 
second.  Furthermore,  loading  the  standard  fork  slightly  with  wax  was 
found  to  decrease  the  number  of  beats.  Hence  the  correct  rate  of  the 
fork  was  508.  Therefore,  since  S  =  nX,  the  observed  value  of  5  was 
1016  x  34.0  =  34,544  cm.,  a  value  which  differs  from  the  theoretical  result 
by  but  .07  per  cent. 

When  the  fork  was  held  3  cm.  above  the  end  of  the  pipe  the  first  reso- 
nant length  was  found  to  be  15.7  cm.  Hence  x  =  17  —  15.7  =  1.3  cm.  The 
radius  of  the  pipe  was  1.8  cm.  Hence  this  correction  was  .71  of  the  radius. 


CHAPTEE   XVIII 
THE  MUSICAL  PROPERTIES  OF  AIR  CHAMBERS 

129.  Notes  to  which  a  closed  pipe  will  respond.    It  has  been 
shown  in  section  128  that  resonance  is  possible  in  a  pipe  closed 
at  one  end  when  the  corrected  length  of  the  pipe  (which  will  be 
denoted  by  P)  is  an  odd  multiple  of  one  fourth  of  the  wave  length  X 
of  the  train  of  waves  entering  the  pipe  ;  i.e.  when  P  =  \  X,  or  when 
P  =  I  X,  or  when  P  =  |  X,  and  so  on.    If  the  pipe  length  is  kept 
constant  while  the  wave  length  is  changed,  it  follows  at  once  that 
a  succession  of  wave  lengths,  \,  X2,  X3,  X4,  and  so  on,  will  be 
found  to  which  the  pipe  will  respond,  and  that  Xj  =  4  P,  X2  =  ^  P, 
X3  =  |  P,  and  X4  =  ^  P,  etc.    These  wave  lengths  bear  the  ratios 
1,  1,  l,  l,  etc.    Hence,  since  vibration  numbers,  n,  vary  inversely 
as  wave  lengths  (for  S  =  n\),  the  vibration  numbers  of  the  notes 
which  are  able  to  produce  resonance  in  a  closed  pipe  must  bear 
the  ratios  1,  3,  5,  7,  9,  and  so  on.    The  note  of  longest  wave  length 
to  which  a  given  pipe  can  respond  is  called  the  fundamental  of 
the  pipe ;  the  notes  of  higher  frequency  which  give  resonance  are 
called  its  overtones. 

130.  Notes  to  which  an  open  pipe  will  respond.    If  it  is  true,  as 
stated  in  section  128,  that  a  wave  which  is  traveling  down  a  pipe 
is  reflected,  upon  reaching  an  open  end,  just  as  it  would  be  if  it 
had  come  to  a  new  medium  of  smaller  density  than  that  within 
the  pipe,  then  it  ought  to  be  possible  to  obtain  the  phenomenon  of 
resonance  with  open  as  well  as  with  closed  pipes  ;  and,  furthermore, 
the  shortest  length  of  an   open  pipe  which  should  produce  the 
reenforcement  of  a  given  train  of  waves  should  be  twice  as  great 
as  the  shortest  resonant  length  of  a  closed  pipe.    For,  if  a  pulse 
of  condensation  0  (Fig.  142),  in  which  the  particles  are  moving 
from  left  to  right,  is  reflected  upon  reaching  B  as  a  pulse  of  rare- 
faction, i.e.  as  a  motion  of  the  particles  of  the  returning  wave 

201 


202 


ELECTRICITY,  SOUND,  AND  LIGHT 


from  left  to  right,  it  is  at  once  evident  that  the  shortest  length  of 
AB  for  which  there  can  be  a  union  of  the  direct  and  reflected 
waves  in  the  same  phase  is  that  which  permits  0  to  return  to  A 
just  in  time  to  unite  with  1.  This  means  obviously  that  the  shortest 
resonant  length  must  ~be  one  half  the-  wave  length  of  the  train,  instead 
of  one  fourth,  as  in  the  case  of  a  closed  pipe.  The  fact  that  a 
length  of  open  pipe  can  indeed  always  be  found  which  will 
respond  just  as'  loudly  to  a  given  note  as  any  closed  pipe,  and 
that  this  length  is  twice  as  great  as  that  of  the  shortest  resonant 
closed  pipe,  may  be  taken  as  complete  experimental  demonstra- 
tion of  the  statement  made  in  section  128  as  to  the  nature  of  the 
reflection  occurring  when  a  wave  reaches  the  open  end  of  a  pipe. 
If  the  open  pipe  is  gradually  lengthened,  there  should  obviously 
again  be  resonance  when  0  returns  to  A  just  in  time  to  unite  with 


FIG.  142 


2  ;  i.e.  when  the  pipe  length  has  been  increased  by  one  half  wave 
length,  and  again  when  0  returns  in  time  to  unite  with  3,  etc.  In 
a  word,  an  open  pipe  should  produce  resonance  when  its  length  is 
a,ny  multiple  whatever  of  J-X;  i.e.  when  the  pipe  length  bears  to  the 
wave  length  any  of  the  ratios  \,  f ,  |,  |,  |,  f,  etc.  This  is  equiva- 
lent to  saying  that  the  notes  which  will  produce  resonance  in  a 
given  open  pipe  of  fixed  length  must  bear  the  frequency  ratios 
1,  2,  3,  4,  5,  etc.  In  other  words,  while  the  fundamental  and  over- 
tones of  a  pipe  closed  at  one  end  must  bear  the  frequency  ratios 
represented  by  the  odd  numbers  only,  the  fundamental  and  over- 
tones of  an  open  pipe  should  bear  the  frequency  ratios  represented 
by  all  the  numbers,  even  and  odd.  This  is  often  stated  thus :  In 
closed  pipes  only  the  odd  overtones  are  possible  ;  in  open  pipes  all 
the  overtones,  even  and  odd,  are  possible. 


MUSICAL  PROPERTIES  OF  AIR  CHAMBERS      203 

131.  Natural  periods  of  pipes.  Not  only  will  a  given  pipe,  open 
or  closed,  intensify,  as  explained  above,  trains  of  waves  of  certain 
definite  wave  length  which  present  themselves  at  its  mouth,  but 
a  single  pulse  entering  such  a  pipe  must  be  returned,  by  virtue  of 
successive  reflections  at  the  ends,  as  a  succession  of  pulses  follow- 
ing one  another  at  equal  intervals.  In  other  words,  a  single  pulse 
must  be  given  back  by  the  pipe  as  a  musical  note,  of  very  rapidly 
diminishing  intensity,  it  is  true,  but  of  perfectly  definite  wave 
length.  Furthermore,  this  wave  length  must  be  the  wave  length 
of  the  train  which  is  capable  of  producing  the  fundamental  reso- 
nance of  the  pipe.  For,  if  the  pipe  is  closed,  for  example,  at  the 
lower  end,  then  the  first  time  the  pulse  returns  to  the  mouth  after 
reflection  at  the  closed  end  it  will  produce  an  outward  motion  of 
the  particles  near  the  mouth,  the  next  time  an  inward  motion,  the 
next  time  an  outward  motion,  and  so  on  ;  i.e.  the  pulse  must  travel 
four  times  the  length  of  the  pipe  in  the 
interval  between  the  appearance  of  two 
successive  condensations  at  the  mouth. 
The  length  of  the  pipe  is  thus  one  fourth 


of  the  wave  length  of  the  note  given  off  FlG  143 

by  it,  and  this  is  the  relation  which  ex- 
ists in  the  case  of  a  train  of  waves  producing  the  fundamental 
resonance.    The  pipe  is  therefore  said  to  have  a  natural  period, 
or  to  be  capable  of  producing  a  note  of  wave  length  four  times  as 
great  as  its  own  length. 

If  the  pipe  is  open  instead  of  closed  at  the  farther  end,  a  single 
pulse  (a  condensation)  entering  at  a  (Fig.  143)  will  emerge  at  b  first 
as  a  motion  of  the  particles  from  left  to  right.  The  reflected  portion 
will  then  travel  back  through  the  tube  as  a  motion  of  the  particles 
from  left  to  right  (a  rarefaction),  which  will  in  turn  be  reflected 
at  a,  still  as  a  motion  from  left  to  right ;  and  thus,  after  traveling 
the  length  of  the  tube  twice,  the  pulse  will  again  emerge  at  5  in 
its  original  direction  (a  condensation).  Thus,  in  this  case,  the  wave 
length  of  the  train  of  waves  into  which  the  pipe  has  transformed 
the  single  pulse  is  twice  the  length  of  the  pipe ;  i.e.  the  note  given 
off  by  the  pipe  has,  as  before,  the  same  wave  length  as  that  which 
will  produce  the  fundamental  resonance  in  the  pipe.  This  note  is, 


204 


ELECTRICITY,  SOUND,  AND  LIGHT 


of  course,  an  octave  higher  than  the  note  given  off  by  a  closed 
pipe  of  the  same  length.  It  is  this  ability  of  a  pipe,  open  or  closed, 
to  pick  up  irregular  pulses  and  transmute  them  by  successive 
reflections  into  notes  of  definite  pitch  which  explains  the  contin- 
uous humming  in  definite  pitch  which  is  heard  when  a  tube,  a  sea- 
shell,  or  any  sort  of  cavity  of  sufficient  size  is  held  close  to  the  ear. 
132.  Production  of  the  fundamentals  of  pipes  by  air  jets.  In 
order,  however,  that  a  pipe  may  be  made  to  give  forth  its  funda- 
mental note  distinctly,  it  is  necessary  to  do  more  than  to  start  a 

single  pulse  in  at  one  end ;  for  the  energy 
of  this  pulse  is  dissipated  so  rapidly  in  the 
successive  reflections  and  transmissions 
that  only  when  the  pipe  is  placed  very 
close    to    the    ear   can   anything  which   resembles   a 
musical  note  be  recognized  at  all.    If,  however,  a  gen- 
*  tie  current  of  air  is  directed  continuously  against  one 

edge  of  the  pipe,  as  in  Figure  144,  the  fundamental 
note  can  be  made,  with  suitable  blowing,  to  come  out 
very  strongly.  In  order  to  understand  this  action, 
consider  first  a  pipe  closed  at  the  lower  end,  and  sup- 
pose that  the  original  current  of  air  is  so  directed  as 
to  strike  at  a  just  inside  the  edge  (see  Fig.  144).  A 
condensation  starts  down  the  pipe  and  is  reflected, 
when  it  reaches  the  bottom,  as  a  condensation,  i.e.  as 
an  upward  motion  of  the  particles.  When  this  con- 
densation reaches  the  mouth  it  pushes  the  current  of 
air  outside  of  the  edge.  This  starts  a  rarefaction  down 
the  pipe  which,  upon  its  return  to  the  mouth  as  a  rarefaction, 
draws  the  current  of  air  inside  the  edge  again.  Thus  the  current 
is  made  to  vibrate  back  and  forth  over  the  edge,  the  period  of 
its  vibration  being  controlled  entirely  by  the  natural  period  of  the 
pipe;  for  between  two  instants  of  emergence  of  the  jet  from  the 
pipe  a  rarefaction  must  travel  twice  the  length  of  the  pipe  and 
then  a  condensation  must  do  the  same ;  i.e.  a  sound  pulse  must 
travel  four  times  the  length  of  the  pipe.  Hence  the  wave  length 
of  the  emitted  note  is  the  same  as  that  which  corresponds  to  the 
natural  period;  i.e.  it  is  four  times  the  length  of  the  pipe.  The 


FIG.  144 


MUSICAL  PROPERTIES  OF  AIK   CHAMBERS      205 

source  of  the  musical  note  is  to  be  found,  then,  in  the  vibration  of 
the  air  jet  into  and  out  of  the  end  of  the  pipe.  The  pipe  itself 
may  be  looked  upon  merely  as  a  device  for  enabling  the  jet  to 
send  pulses  to  the  ear  with  perfect  regularity. 

The  theory  of  the  open  pipe  differs  only  slightly  from  that  of 
the  closed.  If  the  jet  is  directed  just  inside  the  edge,  a  condensa- 
tion starts  down  the  pipe,  and  at  the  same  time,  as  is  indeed  also 
the  case  with  the  closed  pipe,  the  pressure  within  the  upper  end 
of  the  pipe  begins  to  rise  because  of  the  influx  of  air.  If  the  blow- 
ing is  of  just  the  right  intensity,  this  pressure  may  force  the  jet 
outside  the  edge  at  just  the  instant  at  which  the  original  conden- 
sation reaches  the  lower  end  and  starts  back,  in  this  case  as  a 
rarefaction.  When  this  returning  rarefaction  reaches  the  mouth 
it  draws  the  jet  inside  again.  At  this  instant  the  rarefaction  which 
started  down  the  pipe  when  the  jet  first  swung  outside  has  just 
reached  the  lower  end  of  the  tube.  Upon  its  return  to  the  mouth 
as  a  condensation  it  drives  the  jet  again  outside,  and  thus  the  jet 
is  alternately  forced  back  and  forth  over  the  edge,  its  period  being 
controlled  entirely  by  the  natural  period  of  the  pipe,  for  it  will  be 
seen  that  between  two  successive  emergences  of  the  jet  from  the 
mouth  of  the  tube  a  sound  pulse  travels  down  the  tube  and  back. 
If  the  blowing  is  not  of  just  the  right  intensity  so  that  the  pres- 
sure reaction  near  the  mouth  throws  the  jet  out  for  the  first  time 
at  just  the  instant  at  which  the  first  condensation  reaches  the 
lower  end,  then  the  pulses  reflected  from  the  lower  end  do  not 
reach  the  mouth  at  the  right  instants  to  set  up  regular  vibration 
of  the  jet  over  the  edge,  and  consequently  no  note  is  produced. 

133.  Production  of  the  overtones  of  pipes  by  air  jets.  If,  in 
the  case  of  the  open  pipe,  the  violence  of  the  blowing  is  increased 
to  just  the  right  amount,  the  pressure  within  the  top  of  the  pipe 
may  be  increased  so  rapidly  that  the  jet  is  thrown  out  in  just  one 
half  its  former  period.  In  this  case  the  reflected  pulses  will  get 
back  to  the  mouth  in  just  the  right  time  to  keep  the  vibration 
going,  but  the  note  given  forth  will  be  the  first  overtone  of  the 
open  pipe,  namely  the  octave  of  the  fundamental.  Similarly,  still 
harder  blowing  of  just  the  right  intensity  will  cause  the  jet  to 
swing  out  in  just  one  third  its  former  period,  and  the  returning 


206 


ELECTRICITY,  SOUND,  AND  LIGHT 


pulses  will  then  get  back  to  the  mouth  in  just  the  time  to  keep 
the  jet  vibrating  in  the  period  of  the  second  overtone,  the  fre- 
quency of  which  is  three  times  that  of  the  fundamental,  etc. 
Blowing  of  intermediate  intensities  will  produce 
no  notes  at  all,  since  the  times  of  return  of  the 
reflected  pulses  are  then  such  as  to  interfere 
with  the  period  of  vibration  which  is  starting, 
instead  of  to  keep  it  going. 

The  production  of  overtones  in  closed  pipes  is 
precisely  similar,  save  that  in  order  to  produce 
the  first  overtone  the  blowing  must  be  so  hard 
as  to  cause  the  jet  to  swing  out  of  the  pipe  in 
one  third  of  the  time  required  for  the  first  con- 
densation to  travel  to  the  bottom  and  back,  for 
the  first  overtone  of  a  closed  pipe  has  a  fre- 
quency three  times  that  of  the  fundamental, 
the  second  five  times,  etc.  (see  sect.  129).  By 
blowing  with  varying  degrees  of  violence  across 
either  open  or  closed  tubes,  it  is  generally  easy 
to  produce  three  or  four  notes  of  different  pitch 
which  are  found  to  have  precisely  the  frequen- 
cies demanded  by  the  above  theory.  If  the  pipe 
is  long  and  narrow,  it  may  be  quite  impossible  to 
produce  the  fundamental  for  the  reason  that  the 
jet  is  forced  out  by  the  increased  pressure  long 
before  the  first  pulse  returns  from  the  remote  end. 
134.  Types  of  wind  instruments.  The  above 
theory  explains  the  action  of  nearly  all  wind  in- 
struments. In  organ  pipes  (Fig.  145)  the  current 
of  air  is  forced  through  the  tube  ab  into  the  air* 
chest  C,  thence  through  the  narrow  slit  de,  into 
the  embrocliure  E  or  mouth  of  the  pipe,  where  it 
passes  as  a  narrow  jet  toward  the  thin  edge  or 
lip  fg.  As  a  result  of  small  differences  in  pressure  inside  and 
outside  of  the  embrochure,  the  jet  is  caused  to  deviate  to  one  side 
or  the  other  of  the  lip.  When  the  pipe  is  sounding,  it  vibrates 
back  and  forth  across  the  lip  precisely  as  the  air  jet  vibrated 


FIG.  145 


MUSICAL  PROPERTIES  OF  AIR  CHAMBERS      207 


FIG.  146 


back  and  forth  across  the  edge  of  the  pipe  in  the  discussion  of 
section  133. 

Flutes  and  whistles  of  all  sorts  are  precisely  similar  in  their 
action  to  organ  pipes.  In  any  of  them  the  air  chamber  may 
be  either  open  or  closed.  In  flutes  it  is  open ;  in 
whistles  it  is  usually  closed ;  in  organ  pipes  it  is 
sometimes  open  and  sometimes  closed.  In  pipe 
organs  there  is  a  different  pipe  for  every  note,  but 
in  flutes,  fifes,  etc.,  a  single  tube  is  made  to  produce 
a  whole  series  of  notes  either  by  blowing  over- 
tones or  by  opening  holes  in  the  side,  —  an  opera- 
tion which  is  equivalent  to  cutting  off  the  tube  at 
the  hole,  since  a  reflected  wave  starts  back  as  soon 
as  a  point  is  reached  at  which  there  is  greater  free- 
dom of  expansion  than  has  been  met  with  before. 

In  the  case  of  some  instruments,  like  the  clari- 
net (Fig.  146),  the  end  of  the  pipe  against  which 
the  performer  blows  is  almost  closed  by  a  reed  I 
which  is  loosely  pivoted  at  the  base  and  free  to  swing,  under  the 
influence  of  an  outside  pressure,  so  as  to  close  the  opening  entirely. 
When  the  performer  blows  upon  this  mouth- 
piece, a  pulse  of  condensation  enters  the  tube 
and  at  the  same  time  the  reed  closes  the  open- 
ing. This  pulse  after  reflection  from  the  open 
end  of  the  clarinet  as  a  rarefaction,  and  a  sub- 
sequent reflection  at  the  mouthpiece  (closed  by 
the  reed),  also  as  a  rarefaction,  is  again  reflected 
at  the  open  end,  but  now  as  a  condensation ; 
and  therefore,  after  traveling  the  tube  four  times, 
the  original  condensation  returns  and  forces  the 
reed  open,  admitting  a  new  pulse.  The  overtones 
which  may  be  produced  in  such  an  instrument 
are  evidently  those  of  a  closed  pipe.  It  is  evi- 
dent that  the  vibration  frequency  is  independent  of  the  reed  and 
depends  only  upon  the  effective  length  of  the  clarinet. 

In  the  case  of  instruments  like  the  trumpet  and  other  brass 
wind  instruments  the  current  of  air  enters  a  mouthpiece  similar 


FIG.  147 


208  ELECTEICITY,  SOUND,  AND  LIGHT 

to  that  shown  in  Figure  147.  The  lips  of  the  performer  act  as  a 
double  reed.  A  pulse  of  condensation  enters,  the  lips  closing  when 
the  reaction  of  its  pressure  equals  that  of  the  air  in  the  rnouth  of 
the  performer.  This  pulse,  reflected  as  a  rarefaction  from  the  open 
end  of  the  trumpet  to  the  lips,  reduces  the  pressure  at  that  point 
and  a  new  pulse  enters.  The  fundamental  depends  then  only  upon 
the  length  of  the  instrument.  The  overtones  are  produced  exactly 
as  in  an  organ  pipe,  by  blowing  more  suddenly  and  to  some  extent 
by  increasing  the  tension  of  the  lips.  The  possible  overtones  are 
those  of  an  open  pipe. 

135.  The  musical  scale.  The  physical  basis  of  harmony  in 
music  lies  in  the  simplicity  of  the  ratios  of  the  vibration  frequen- 
cies of  the  notes  which  are  sounded  together.  When  a  note  and 
its  octave  are  sounded  together  the  result  is  recognized  by  the  ear 
as  agreeable,  and  the  two  notes  are  said  to  be  consonant.  The 
ratio  of  the  frequencies  of  the  two  notes,  known  as  the  interval 
between  them,  is  in  this  case  ^.  The  next  most  consonant  interval 
is  that  between  do  and  sol  (C  and  G).  It  is  found  to  be  the  next 
most  simple  physical  interval,  having  a  value  |.  It  is  known  as  a 
fifth  because  G  is  the  fifth  note  above  C  in  the  sequence  of  eight 
notes  which  constitutes  the  octave  of  the  ordinary  musical  scale. 
If  the  note  G  and  the  octave  of  C  are  sounded  together,  the  inter- 
val is  |,  as  is  evident  from  the  fact  that  |  of  1  is  |  of  2.  These 
three  notes  —  do,  sol,  and  the  octave  of  do  —  thus  utilize  the  three 
simplest  frequency  ratios,  namely  -|,  |,  and  -|.  The  next  most  conso- 
nant interval  is  that  between  do  and  mi  (C  and  E).  It  is  known 
as  the  third  and  represents  the  vibration  ratio  |.  The  notes  do, 
mi,  sol  (C,  E,  G)  sounded  together  are  known  as  the  major  chord. 
It  will  be  seen  from  the  above  that  their  relative  vibration  fre- 
quencies are  as  4  :  5  :  6. 

The  so-called  major  diatonic  scale  is  made  up  of  three  major 
chords.  The  absolute  vibration  number  taken  as  the  starting  point 
is  wholly  immaterial,  but  the  explanation  of  the  origin  of  the 
eight  notes  of  the  octave,  commonly  designated  by  the  letters  C, 
D,  E,  F,  G,  A,  B,  C/  may  be  made  more  simple  if  we  begin  with 
a  note  of  vibration  number  24.  The  first  major  chord  —  do,  mi, 
sol,  or  C,  E,  G  —  would  then  correspond  to  the  vibration  numbers 


MUSICAL   PROPERTIES  OF  AIR   CHAMBERS      209 

24,  30,  36,  or,  as  explained  above,  the  vibration  ratios  4,  5,  6.  The 
second  major  chord  starts  with  C',  the  octave  of  C,  and  comes  down 
in  the  ratios  6,  5,  4.  The  corresponding  vibration  numbers  are 
48,  40,  32,  the  corresponding  notes  do',  la,  fa,  and  the  correspond- 
ing letters  C',  A,  F.  The  third  chord  starts  with  G  as  the  first 
note  and  runs  up  in  the  ratios  4,  5,  6.  This  gives  the  vibration 
numbers  36,  45,  54,  the  syllables  sol,  si,  re,  and  the  letters  G,  B,  D'. 
Since  the  note  D'  does  not  fall  within  the  octave,  its  vibration 
number  being  above  48,  the  note  D,  an  octave  lower  and  having 
a  vibration  number  27,  is  taken  to  complete  the  eight  notes  of 
the  major  diatonic  scale.  The  chord  do-mi-sol  is  called  the  tonic, 
sol-si-re  the  dominant,  and  fa-la-do  the  subdominant.  The  relations 
between  the  notes  of  the  octave  are  given  below  in  tabular  form. 

Notes C  D  E  F  G  A  B  C' 

Frequencies 24  27  30  32  36  40  45  48 

Intervals  with  C   .     .     .      1  f  J  §  f  §  -^  2 

major      major      fourth      fifth      major      seventh      octave 
Name  of  interval      .     .  .     , 

second      third  sixth 

Intervals  with  next  note  f  *£  if  f  -V  f  Jf 

major      minor       half 
Name  of  interval.     .     . 


Any  scale  the  notes  of  which  are  separated  by  these  intervals 
is  known  as  a  scale  of  just  temperament.  The  scale  adopted  by 
physicists  starts  with  middle  C  =  256.  That  adopted  internation- 
ally for  musical  purposes  has  A  =  435.  This  gives  a  series  of 
values  slightly  higher  than  that  of  the  physical  scale.  The  scale 
formed  on  C  =  256  follows. 

Absolute  names   .     .     .    C      D     E        F  G       A  B  C'     D'  E'  F'  G'  A'    B'    C" 

Syllables do     re    mi     fa  sol      la  si  do     re  mi  fa  sol  la    si     do 

Relative  frequencies   .Iff        §  §         §  -^      2       f  \  f  3  Af-    -1,5-      4 

Absolute  frequencies       256   288   320  341.3  384  426.6  480  512   576  640  683  768  853  960  1024 

It  is  often  desirable  in  music  to  adopt  some  other  note  than  C 
as  the  fundamental  note  in  the  scale,  i.e.  as  the  keynote.  Thus  the 
scale  formed  on  G  as  a  fundamental  and  having  intervals  of  just 
temperament  would  be 

do      re     mi     fa      sol      la      si      do 
384     432     480     512     576     640     720     768 


210  ELECTEICITY,  SOUND,  AND  LIGHT 

If,  then,  a  piece  of  music  is  to  be  played  in  just  temperament  in  the 
scale  of  G,  the  instrument  used  must  be  capable  of  producing  two 
notes  in  addition  to  those  found  in  the  major  scale  of  C,  namely 
the  notes  corresponding  to  the  vibration  numbers  432  and  720. 
Similar  investigation  of  the  other  possible  scales  of  A,  B,  D,  E,  and 
F  show  occasion  for  many  other  notes,  so  that  a  piano  which  could 
be  played  in  just  temperament  in  all  the  keys  demanded  by  modern 
music  would  require  about  50  notes  in  each  octave.  Since,  how- 
ever, the  introduction  of  all  these  notes  would  make  the  manipu- 
lation of  the  instrument  a  physical  impossibility,  there  has  been 
devised,  for  keyed  instruments  of  the  piano  type,  another  scale 
which  is  known  as  the  scale  of  even  temperament.  The  origin  of 
this  scale  may  be  seen  as  follows. 

A  careful  comparison  of  all  the  notes  necessary  for  the  various 
scales  (found  as  above  in  the  case  of  G)  shows  that  many  of  these 
notes  differ  so  slightly  that  a  single  note  may  do  satisfactory  duty 
for  several  of  almost  the  same  frequency,  provided  we  are  will- 
ing to  content  ourselves  with  slightly  imperfect  intervals.  Twelve 
notes  are  accordingly  chosen  to  replace  the  fifty,  and  these 
twelve  are,  as  a  matter  of  fact,  made  to  divide  the  octave  into 
twelve  exactly  equal  parts.  It  is  for  this  reason  that  the  scale 
is  called  the  scale  of  even  temperament.  Since  there  are  twelve 
equal  intervals  between  a  note  and  its  octave,  each  interval  is  of 
value  v^,  or  1.059. 

This  scale  is  written  below,  starting  with  C  =  256  for  the  fun- 
damental note.  For  purposes  of  comparison  the  frequencies  of  the 
notes  in  the  scale  of  just  temperament  are  also  given. 

Notation    C    CtforDb     D     DJorEb    E         F   FflorGb     G   GflorAb    A    AJorBb    B      C 
Even  256      271.3      287.4      304.8     322.7    341.7    3G2.2      383.8    40G.G     430.7      456.5    483.5  512 

Just  256  288  320       341  384     -  427  480    512 

It  is  evident  from  the  table  that  although  only  C  and  its  octave 
retain  their  old  values  in  the  scale  of  just  temperament,  the  dif- 
ference between  the  frequencies  of  any  note  in  the  two  scales  is  so 
small  as  not  to  be  in  general  noticeable.* 

*  The  difference  would,  of  course,  be  at  once  noticeable  because  of  the  phe- 
nomenon of  beats  (see  sect.  149,  p.  234),  if  the  same  note  were  sounded  simultane- 
ously in  the  two  scales.  In  general  this  would  not  occur,  for  all  instruments  with 


MUSICAL  PROPERTIES   OF  AIR   CHAMBERS      211 


EXPERIMENT   18 

Object.  To  find  the  overtones  possible  in  open  and  closed  pipes. 

Directions.  Figure  148  shows  an  arrangement  consisting  of  a  rotating 
table  T,  a  siren  W,  and  a  five-foot  open  pipe  P  of  diameter  5  or  6  cm. 
A  current  of  air  is  forced  through  the  glass  tube  g  by  a  large  bellows  or 
other  arrangement.  When  W  is  rotated  the  current  of  air  from  the  bellows 
sends  a  pulse  of  condensation  into  the  pipe  every  time  an  opening  in  W 
comes  over  the  tube  g.  When  W  is  rotated  with  a  frequency 
such  that  the  number  of  these  pulses  entering  the  pipe  is 
equal  to  the  natural  vibration  frequency  of  the  pipe,  resonance 
takes  place  and  the  pipe  gives  forth  loudly  its  fundamental 
note.  In  the  same  way,  if  W  is  rotated  with  the  proper  uni- 
form speed,  P  may  be  made  to  give  forth  loudly  any  desired 
overtone. 

Determine  exactly  the  number  of  holes  which  pass  the  ori- 
fice of  the  pipe  for  one  revolution  of  the  wheel  T.    Cause  the 
pipe  to  give  forth  its  fundamental*  and  hold  the  speed  con- 
stant while  a  second  observer 
takes,  with  a  stop  watch,  the 
time  of  15  revolutions  of  T. 
Measure  the  pipe  length.   From 
the  data  thus  obtained  and  the 
correction  which  it  was  found 
necessary  in  Experiment  17  to 
apply  to  the   open  end  of  a  pipe,  deduce  the  velocity  of   sound  in  air. 

Repeat  the  determination,  using  successively  the  first,  second,  third, 
and  fourth  overtones  instead  of  the  fundamental. 

Transform  P  into  a  closed  pipe  by  inserting  a  cork  in  the  upper  end, 
and  determine  as  above  the  vibration  numbers  corresponding  to  the  fun- 
damental, the  first,  and  the  second  overtones. 

fixed  keyboards  or  frets  are  tuned  to  the  scale  of  even  temperament.  It  is,  how- 
ever, worthy  of  note  that  many  prominent  violinists,  when  playing  without 
accompaniment  by  an  instrument  tuned  in  even  temperament,  instinctively  play 
in  just  temperament  and  so  satisfy  the  demands  of  the  human  ear  for  intervals 
represented  by  the  ratio  of  simple  whole  numbers. 

*  Difficulty  may  be  experienced  in  detecting  the  frequency  corresponding  to 
the  fundamental  because  of  the  fact  that  the  pipe  may  give  off  its  fundamental 
note  weakly  for  a  number  of  pulses  smaller  than  that  corresponding  to  its 
natural  frequency  (see  sect.  131).  This  response  will  scarcely  be  noticeable 
except  when  the  number  of  pulses  is  one  half  that  corresponding  to  the  natural 
period,  when  it  may  be  quite  pronounced.  In  general,  however,  the  funda- 
mental resonance  will  be  so  much  louder  that  it  can  scarcely  be  mistaken.  If 
there  is  doubt,  multiply  at  once  twice  the  pipe  length  by  the  number  of  pulses 
and  see  whether  the  product  is  about  34,000  cm.,  as  of  course  it  should  be. 


FIG.  148 


212  ELECTKICITY,  SOUND,  AND  LIGHT 

EXAMPLE 

The  plate  W  had  40  holes  and  made  9}-|-  revolutions  for  each  revolution 
of  T";  hence  374  pulses  entered  the  tube  P  for  each  revolution  of  T.  The 
pipe  had  a  length  of  113  cm.  and  a  radius  of  3.8  cm.  ;  hence  the  corrected 
length  was  115.6.  The  average  of  three  determinations  of  the  number  of 
seconds  necessary  for  15  revolutions  of  T  when  the  open  pipe  was  emitting 
its  fundamental  note  was  37.4  seconds.  The  average  found  for  30  revo- 
lutions when  the  first  overtone  was  sounding  was  37.2  seconds  ;  for  45 
revolutions  with  the  second  overtone  it  was  37.6  ;  and  for  60  revolutions 
with  the  third  overtone,  37.4.  The  corresponding  frequencies  were  150, 
300,  450,  and  600  ;  that  is,  they  were  in  the  ratio  of  1,  2,  3,  4.  The  aver- 
age speed  of  sound  from  these  values  of  the  frequency  and  pipe  length  was 
347.6  m.  per  second.  The  correct  value  as  found  from  the  temperature 
21. 5°  C.,  and  the  relation  of  section  126  was  344.1  m.  per  second.  The 
difference  was  about  1  per  cent. 

A  cork  placed  in  the  upper  end  of  the  pipe  shortened  its  corrected 
length  to  113.7  cm.  The  average  time  for  15  revolutions  when  the  funda- 
mental of  this  closed  pipe  was  sounding  was  74.3  seconds.  The  time  of  15 
revolutions  with  the  first  overtone  was  24.7  ;  with  the  second  overtone  it 
was  14.8  seconds.  The  corresponding  frequencies  were  75.5,  227,  and  380; 
that  is,  the  frequencies  were  in  the  ratio  of  1,  3,  5.  The  average  value  of 
the  speed  of  sound  from  these  observations  was  344  m.  This  determina- 
tion with  a  closed  pipe  differed  therefore  from  that  made  with  an  open 
pipe  by  1.2  per  cent. 


CHAPTEK   XIX 
LONGITUDINAL  VIBRATIONS  OF  RODS 

136.  Velocity  of  waves  in  thin  rods  of  elastic  material.    The 

analysis  of  section  122  shows  that  if  there  is  no  possibility  of 
lateral  expansion  of  the  medium,  the  velocity  of  a  compressional 
wave  depends  only  upon  the  bulk  modulus  of  elasticity  and  the 
density  of  the  medium.  This  condition  is  realized  when  a  disturb- 
ance originates  in  the  midst  of  an  elastic  medium  of  great  extent 
in  all  directions.  But  when  the  wave  travels  along  a  thin  rod 
there  is  a  slight  lateral  expansion  of  that  portion  of  the  rod  which 
is  undergoing  the  compression.  Hence  if  we  imagine  a  rod  of 
1  sq.  cm.  cross  section  divided  into  centimeter  cubes  after  the 
fashion  of  section  122,  and  if  we  imagine  a  small  pressure  dp 
to  be  applied  by  means  of  a  piston  p  at  one  end  (Fig.  134), 
then  while  each  cube  is  undergoing  the  voluminal  compression 
dv,  the  piston  will  move  forward,  not  now  dv,  but  some  distance 
ds  numerically  a  trifle  larger  than  dv.  From  reasoning  identical 
with  that  given  on  page  188,  an  -equation  results  which  differs 
from  equation  (2),  page  189,  namely  $2  =  dp/pdv,  in  no  respect 
save  that  ds  replaces  dv.  We  obtain,  then, 

dp 


But  ds  is  the  change  in  the  length  of  the  rod  per  unit  length. 
Hence  dp/ds  is  Young's  Modulus  (Y).*  Thus  in  thin  rods  com- 
pressional wavks  move  with  a  velocity  which  is  given  by  the  equation 


*See  "Mechanics,  Molecular  Physics,  and  Heat,  p.  67. 
213 


214 


ELECTRICITY,  SOUND,  AND  LIGHT 


137.  Natural  periods  of  free  rods.    A  rod  surrounded  by  air  is 
in  every  respect  analogous  to  an  open  pipe,  for  the  reflections  at 
the  ends  are  such  as  occur  when  a  wave  passes  from  a  denser  to  a 
rarer  medium.    Thus  such  a  rod  will  respond  to  a  train  of  waves 
if  it  is  of  such  length  that  a  pulse  0  (Fig.  149)  just  entering  the 
rod  at  A  as  a  condensation  will,  after  reflection  at  B,  return  to  A 
and  be  again  reflected  as  a  condensation  at  the  precise  instant  at 
which  pulse  1  reaches  A.     The  length  %AB  is  then  the  distance 
which  a  pulse  0  travels  in  the  rod  before  the  succeeding  pulse  1 
enters  the  rod.    This  is,  by  definition,  one  wave  length  of  the  note 
in  the  rod. 

If  one  single  pulse  strikes  the  rod,  the  successive  reflections 
of  this  pulse  at  A  and  B  will  cause  a  train  of  waves  to  be  given 
off  at  each  end.  Thus  the  rod  will  emit  a  musical  note  the  wave 

length  of  which  in 
the  rod  is  twice  the 
length  of  the  rod. 
The  wave  length 
in  air  of  this  note 
obviously  bears 
the  same  relation  to  its  wave  length  in  the  rod  as  the  velocity 
of  the  wave  in  air  bears  to  its  velocity  in  the  rod. 

If  the  rod  be  clamped  in  the  middle,  it  will  respond  to  and  give 
off  precisely  the  same  note  as  though  it  were  free,  for  the  com- 
pression produced  by  the  clamp  at  the  middle  produces  at  that 
point  the  same  sort  of  a  reflection  as  occurs  at  the  boundary  of  a 
denser  medium ;  hence  the  rod  is  equivalent  to  two  closed  pipes, 
each  of  which  gives  off  the  same  note  as  would  an  open  pipe  (i.e. 
a  free  rod)  of  double  the  length.  In  order  to  set  a  rod  into  longi- 
tudinal vibrations  of  this  sort,  it  is  customary,  instead  of  striking 
one  end,  to  clamp  it  in  the  middle  and  stroke  it  with  a  rosined 
cloth  if  it  is  of  metal,  or  with  a  wet  cloth  if  it  is  of  glass.  The 
mechanism  of  the  tone  production  in  this  case  will  be  more  fully 
discussed  in  section  140. 

138.  Comparison  of  the  velocities  of  sound  in  two  solids.   The 
above  theory  suggests  an  extremely  simple  and  satisfactory  means 
of  comparing  the  velocities  of  sound  in  two  solids.    Thus  we  have 


FIG. 149 


LONGITUDINAL  VIBRATIONS  OF  RODS 


215 


only  to  find  the  vibration  frequencies  (the  pitches)  of  two  notes 
produced  by  stroking  steel  and  brass  rods  of  the  same  length,  in 
order  to  find  the  relative  velocities  of  sound  in  steel  and  brass. 
For  with  rods  of  equal  length  the  number  of  pulses  communicated 
to  the  air  per  second  by  the  traveling  of  pulses  up  and  down  the 
rods  is  obviously  proportional  to  the  velocities  of  sound  in  the  two 
rods.  Thus  if  S8  and  Sb  represent  these  velocities  in  steel  and 
brass  respectively,  and  ns  and  nb  the  corresponding  frequencies 
produced  by  the  rods  of  equal  length,  we  have 


(3) 


The  frequencies  ns  and  nb  can  be  determined  in  a  variety  of  ways; 
for  example,  by  changing  the  length  of  a  given  sonometer  wire  until 
it  is  in  tune,  first,  with  the  note  from  the  steel,  and  then  with 
that  from  the  brass.  Since  the  frequencies  of  the  notes  produced 
under  these  circumstances  are  inversely  proportional  to  the  lengths 
(see  Chap.  XX),  we  have,  if  18  and  lb  are  the  lengths  of  the  same 
wire  which  are  in  tune  with  the  steel  and  brass  respectively, 

s.    i,, 


S,      L 


(4) 


139.  Nodes  and  loops  in  pipes  and  rods.  A  careful  considera- 
tion of  the  resonance  of  pipes  which  are  giving  off  the  first  or 
higher  overtones  reveals  effects  which  have  thus  far  been  over- 
looked. For  example,  it  was  shown  that  when  a  pipe  has  its 


FIG.  150 

second  resonant  length,  a  condensation  0  (Fig.  150)  must  return 
to  the  mouth  of  the  pipe  at  the  instant  at  which  the  rarefaction  | 
reaches  the  mouth;  but,  in  the  return  after  reflection,  0  must 
somewhere  in  the  pipe  collider  with  the  advancing  condensation  1. 
Since  at  the  instant  of  the  reflection  of  0,  1  is  one  wave  length 


216 


ELECTRICITY,  SOUND,  AND  LIGHT 


behind  0,  it  is  evident  that  this  collision  must  take  place  just  | 
wave  length  from  the  end  of  the  pipe,  namely  at  n.  Such  a  col- 
lision of  two  oppositely  moving  condensations  is  entirely  analogous 
to  the  collision  of  two  oppositely  moving  perfectly  elastic  balls. 
These  are  shown  simply  to  exchange  motions,*  the  effect  being 
the  same  as  though  each  ball  passed  through  the  other  without 
experiencing  any  effect  whatever  from  it.  Thus  the  waves  may 
be  thought  of  as  passing  through  one  another,  and  their  mutual 
effects  may  be  ignored.  As  a  matter  of  fact,  it  is  of  course  1  which 
returns  to  the  left  after  the  collision  and  unites  with  f  at  the  mouth, 
while  0  is  forced  back  again  toward  the  closed  end  of  the  pipe. 

One  half  period  after  the  collision  at  n  (Fig.  150)  of  the  con- 
densations 1  and  0  (-^-  -*-)  there  will  occur  at  n  a  collision  of  the 
rarefactions  |-  and  J-  (-*-  -»-).  Thus  the  particles  near  n  are  first 
pushed  together  by  opposing  forces,  then  pulled  apart  by  opposing 
forces.  The  result  is  that  they  do  not  move  at  all.  The  matter 


FIG.  151 

about  n  suffers  alternate  compression  and  expansion,  but  the  par- 
ticles at  n  can  never  move  either  to  left  or  to  right,  because  they 
are  always  being  urged  in  opposite  directions  by  the  oppositely 
moving  waves.  The  .point  n  is  called  a  node.  The  points  between 
the  nodes  where  the  disturbance  is  greatest  are  called  loops. 

If  the  length  of  the  pipe  is  -|,  J,|,  etc.,  wave  length,  it  is  evident 
from  considerations  precisely  like  the  above  that  there  will  be 
nodes  at  n1,  n",  n'",  etc.  (Fig.  151).  In  other  words,  in  any  reso- 
nant closed  pipe  (and  it  is  to  be  remembered  that  such  a  pipe  is 


*  See  "Mechanics,  Molecular  Physics,  and  Heat,"  Chapter  VII. 


LONGITUDINAL  VIBRATIONS  OF  KODS 


217 


resonant  when  and  only  when  its  length  is  an  odd  number  of 
fourth  wave  lengths)  the  first  node  is  one  fourth  wave  length 
from  the  open  end,  and  other  nodes  follow  at  intervals  of  one  half 
wave  length.  The  conventional  method  of  representing  nodes  and 
loops  in  pipes  is  that  used  in  Figure  151. 

Since  the  first  resonant  length  of  an  open  pipe  is  one  half  wave 
length,  and  since  a  condensation  0  is  reflected  as  a  rarefaction,  it 
is  evident  that  0  will  collide  in  the  middle  of  the  pipe  with  J-. 
Hence  an  open  pipe  responding  to  its  fundamental  has  a  node  in 


n' 


FIG.  152 

the  middle.  Similarly  an  open  pipe  responding  to  its  first  over- 
tone has  nodes  at  n1  and  n" ,  each  one  fourth  wave  length  from 
an  end  and  one  half  wave  length  apart.  Similarly  for  the  higher 
overtones  (see  Fig.  152). 

Although  the  preceding  discussion  has  been  limited  to  pipes, 
yet,  since  by  section  137  a  rod  surrounded  by  air  acts  in  every 
respect  like  an  open  pipe,  the  above  conclusions  hold  also  for  rods. 

140.  Kundt's  tube  experiment.  A  rod  mn  (Fig.  153),  to  one 
end  of  which  is  attached  a  light  cork  piston  B,  is  supported  by  a 
clamp  C  at  its  middle  point.  The  piston  B  fits  very  loosely  in  a  long 
tube  AB,  one  end  of  which  is  closed  by  a  tightly  fitting  piston  A. 

The  rod  mn  is  set  into  longitudinal  vibration  by  drawing  along 
it  a  cloth  covered  with  rosin.  It  would  seem  at  first  thought  as 
though  the  slipping  of  the  cloth  along  the  rod  were  so  irregular 
that  no  musical  note  could  be  produced.  As  a  matter  of  fact, 
however,  the  slipping  is  controlled  by  the  natural  period  of  the 
rod  in  much  the  same  way  as  the  vibrations  of  the  air  jet  at  the 


218  ELECTRICITY,  SOUND,  AND  LIGHT 

mouth  of  an  organ  pipe*  are  controlled  by  the  natural  period  of 
the  pipe.  Thus  the  first  slip  starts  a  pulse  down  the  rod  which, 
because  of  the  reflections  at  the  ends,  returns  to  the  starting  point 
at  stated  intervals.  Of  course  the  tendency  to  slip  is  greatest 
at  the  instant  of  the  return  of  the  first  pulse,  so  that  succeeding 
slips  take  place  at  the  instants  of  return  of  succeeding  pulses. 
Thus  the  rod  gives  off  loudly  the  note  corresponding  to  its  natu- 
ral period.  The  rod  mn  is  clamped  in  the  middle,  but  it  was 
shown  above  f  that  the  natural  period  in  this  case  is  precisely 
the  same  as  when  the  rod  is  free.  The  wave  length  of  the  note 
produced  in  the  material  of  which  the  rod  is  composed  is  there- 
fore twice  the  length  of  the  rod. 

The  piston  A  is  so  adjusted  in  position  that  the  air  column  AB 
is  of  such  a  length  as  to  be  resonant  to  the  note  of  the  rod.  Nodes 
are  then  formed  which  may  be  brought  into  evidence  by  placing 


FIG. 153 

along  the  bottom  of  the  tube  a  layer  of  light  cork  filings.  The 
cork  dust  will  be  collected  into  ridges  at  the  points  of  maximum 
disturbance,  i.e.  at  the  loops.  The  explanation  of  the  fact  that  each 
loop  is  marked  not  by  a  single  ridge,  but  by  a  series  of  ridges, 
demands  an  analysis  which  is  beyond  the  scope  of  this  text. 

The  wave  length  of  the  note  given  off  by  the  rod  may  be  obtained 
at  once  by  measuring  the  distance  between  two  successive  loops. 
If,  then,  Sr  represents  the  velocity  of  the  wave  in  the  rod,  \r  its 
wave  length  in  the  rod,  and  n  the  frequency,  we  have 

Sr  =  n\r. 

Also,  if  the  tube  contains  air,  and  S  represents  the  velocity  of 
sound  waves  in  air,  and  X  the  wave  length  of  a  note  of  frequency 

*  See  section  132,  page  204. 
t  See  section  137,  page  214. 


LONGITUDINAL  VIBRATIONS  OF  RODS  219 

7i,  then  *S'  =  n\.  Combining  these  two  equations  we  obtain  as  the 
expression  for  the  velocity  of  compression al  waves  in  the  material 
of  the  rod,  ~ 

Sr  =  ^S,  (5) 

in  which  Xr  is  merely  2  mn  and  X  is  twice  the  distance  between 
nodes  in  air. 

By  similar  reasoning,  if  some  other  gas  is  substituted  for  the 
air  in  the  tube  and  Sg  and  X^,  represent  the  velocity  and  wave 
length  of  the  same  note  in  that  gas,  then  Sg  =  n\g>  or 

c<  \<7    o  /f\ 

,S,  =  -A.  (6) 

Since  S  has  already  been  determined,  and  X  and  X^  may  be  ob- 
served, the  velocity  of  sound  in  any  gas  may  be  found. 


EXPERIMENT  19 

(A)  Object.  To  find  the  velocity  of  compressional  waves  in  steel. 

Directions.  Following  the  method  of  the  Kundt's  tube  experiment  de- 
scribed in  section  140,  adjust  carefully  the  sliding  piston  A  (Fig.  153) 
until  a  maximum  of  agitation  of  the  cork  dust  at  the  loops  is  produced 
when  a  steel  rod  mn  is  stroked  with  a  rosined  cloth.  Measure  the  distance 
between  A,  which  is  a  node,  and  the  node  most  remote  from  A,  say  the 
nth,  then  that  between  A  and  the  (n  —  l)st  node,  then  between  A  and  the 
(n  —  2)d,  etc.  Make  two  vertical  columns,  one  of  measured  distances, 
the  other  of  the  corresponding  numbers  of  half  wave  lengths.  The  sum 
of  the  first  column  divided  by  the  sum  of  the  second  gives  the  most 
accurate  value  of  -|  wave  length  which  is  obtainable  from  this  sort  of  an 
observation.  Shake  up  the  cork  dust  and  obtain  a  second  set  of  readings, 
measuring  this  time  first  between  the  first  and  last  loops,  then  -between 
the  second  and  next  to  the  last,  etc.,  and  averaging  as  described  above. 
If  Sr  represents  the  velocity  of  sound  in  steel,  Xr  the  wave  length  in  steel 
of  the  note  produced,  and  if  S  and  X  represent  the  corresponding  quantities 
when  the  waves  have  passed  over  into  air,  then  Sr  can  be  found  at  once 
from  the  relation  Sr  =  (\r/\)S.  Then  from  Sr  and  Xr  the  frequency  n  may 
be  obtained  if  desired. 

Compare  the  observed  value  of  the  velocity  in  steel  with  the  theoret- 
ical value  deduced  from  Young's  Modulus,  and  the  density,  by  use  of  the 
relation  S  =  ^Y. 


220  ELECTRICITY,  SOUND,  AND  LIGHT 

(B)  Object.  To  find  the  velocity  of  sound  in  CO2. 

Directions.  Replace  the  air  in  the  Kuudt  tube  by  CO2  by  permitting  a 
gentle  current  from  a  charged  cylinder  to  pass  in  at  o  (Fig.  153)  and  out 
at  o'  for  two  or  three  minutes.  Then  determine,  precisely  as  above,  the 
wave  length  in  CO2  of  the  note  given  forth  by  the  steel  rod.  Thence 
deduce  the  velocity  of  sound  in  CO2  and  compare  the  result  with  the  theo- 
retical value  obtained  from  the  barometer  height,  the  density  of  CO2 
(viz.  1.53  x  the  density  of  air),  and  the  value  of  y  (=  1.30). 

(C)  Object.   To  compare  the  velocities  of  sound  in  brass  and  steel. 
Directions.  Take  two  equal  rods  three  or  four  meters  long,  one  of  steel 

and  one  of  brass.  Set  them  successively  into  longitudinal  vibrations  by 
clamping  them  in  the  middle  and  stroking  with  a  rosined  cloth.  By  means 
of  a  sliding  bridge  vary  the  length  of  a  small  wire  of  a  sonometer  (or 
violin)  until,  when  picked  transversely,  it  produces  first  a  note  in  tune  with 
that  of  the  steel  rod,  then  with  that  of  the  brass  rod.  Determine  the  rela- 
tive velocities  by  the  method  of  section  138  and  compare  with  the  relative 
values  obtained  by  taking  the  values  of  Young's  Modulus  and  the  appro- 
priate densities  from  a  table  and  substituting  in  equation  (2),  page  213. 


EXAMPLE 

(A)  The  observations  of  the  distance  between  nodes  made  according  to 
the  directions  given  above  were  as  follows. 

Number  of  i  Number  of  A 

Distance  in  cm.  Distance  in  cm. 

wave  lengths  wave  lengths' 

7  65.3                                    6                          56.1 

6  55.7                                    4                         36.5 

5  47.5                                  _2                          17.7 

4  38.0                                  12                        110.3 

3  28.5                                        .-.IX  =9.2  cm. 

2  19.0 

J.  9.5 

28  263.5 

.-.  IX  =  9.4  cm. 

The  average  value  of  A  in  air  was  thus  found  to  be  18.6  cm.  The  tem- 
perature of  the  room  was  26° C.  The  velocity  of  sound  in  air  at  this 
temperature  is  331.27  +  26  x  .6  =  346.9m.  per  second.  The  length  mn 
was  136.5cm.,  thus  making  \r  =  273cm. 

°73 

Hence  Sr  for  steel  was  ^— -  •  346.9  =  5073  m.  per  second. 
18. o 

Using  the  value  of  Young's  Modulus  previously  found  from  experiments 
on  a  steel  wire,  namely  19.7  x  1011,  and  taking  7.8  as  the  density  of  steel, 


LONGITUDINAL  VIBRATIONS  OF  RODS          221 

the  theoretical  value  of  the  velocity  of  sound  in  steel,  as  calculated  from 
the  formula  S  —  A/—  was  found  to  be  5026  m.  per  second.  The  difference 

between  the  two  values  is  .9  per  cent. 

(B)  When  the  tube  was  filled  with  CO2  at  atmospheric  temperature  the 
average  wave  length  produced  in  the  CO2  by  the  vibrations  of  the  steel 

rod  was  14.5.    Hence  the  velocity  in  CO2  was  346.9  x  ii^  =  270.4  m.    The 

18.6 

corrected  barometric  pressure  was  74.22  cm.  The  value  of  the  velocity 
calculated  from  this  pressure,  together  with  the  density  of  carbon  dioxide, 
namely  .001152  x  1.5:3,  and  the  factor  y  =  1.30,  was  found  to  be  269.4. 
The  difference  was  thus  .4  per  cent. 

(C)  Rods  of  steel   and  brass   290  cm.   long  were  tuned  to  lengths  of 
21.2  cm.  and  30.5  cm.  respectively  on  a  given  sonometer  wire. 

Hence  |  ' 

The  theoretical  value  obtained  by  using  7.8  and  8.4  respectively  for  the 
densities  of  steel  and  brass,  and  19.7  x  1011  and  10.2  x  1011  for  the  corre- 
sponding values  of  Young's  Modulus,  was  1.442,  a  difference  of  0.3 
per  cent. 


CHAPTER   XX 
WAVES   IN   STRINGS 

141.  General  characteristics  of  wave  motion.  In  the  preceding 
sections  we  have  discussed  only  compressional,  or  longitudinal, 
wave  motion,  and  have  found  this  to  be  characterized  by  the  fact 
that  the  particles  which  transmit  the  wave  move  in  the  line  of 
propagation  of  the  wave  itself.  This  is  the  only  sort  of  an  elastic 
wave  which  is  possible  in  substances  which  do  not  possess  rigidity.* 
But  in  substances  which  possess  rigidity  another  type  of  elastic 
wave  motion  is  possible,  namely  transverse  wave  motion.  This  is 
characterized  by  the  fact  that  the  particles  of  the  medium  move 
in  paths  which  are  perpendicular  to  the  direction  of  propagation 
of  the  wave.  The  waves  which  travel  along  a  rope  when  one  end 
is  caused  to  vibrate  by  the  hand  are  of  this  sort. 

Before  this  second  type  is  discussed  it  is  of  importance  to  have 
clearly  in  mind  the  general  characteristics  of  wave  motion.  These 
may  be  seen  from  a  consideration  of  Figure  154,  which  represents 
the  effect  on  the  particles  of  a  medium  of  an  oscillatory  motion 
of  the  piston  P.  The  unit  cubes  of  Figure  134,  section  122,  page 
187,  are  here  replaced  by  vertical  lines.  Thus  Figure  154,  «,  rep- 
resents the  state  of  the  medium  before  the  piston  has  begun  to 
move.  Figure  154,  5,  represents  its  state  when  the  piston  has  un- 
dergone its  greatest  displacement  to  the  right  and  is  ready  to 
return,  a  condition  which  is  represented  by  the  double  arrow (<  »). 
In  the  upper  line  of  arrows  each  arrow  represents  the  direction 
of  displacement  of  the  layer  toward  which  it  points.  The  small 
arrows  below  the  vertical  lines  show  the  direction  of  the  motion 
of  the  layers.  The  zero  below  any  line  indicates  either  that  the 
medium  is  there  at  its  mean  (original)  position,  or  is  just  changing 
the  direction  of  its  motion  at  the  end  of  its  path.  The  succeeding 

*  The  large  waves  on  the  surface  of  a  body  of  water  are  gravity  waves  and 
have  nothing  to  do  with  the  elasticity  of  matter. 

222 


WAVES  IN  STRINGS 


223 


figures  show  the  progression  of  the  initial  condensation  and  the 
subsequent  rarefaction  for  the  indicated  positions  and  directions 
of  motion  of  the  piston.  Obviously  a  condensation  exists,  for  ex- 
ample, in  Figure  154,  6,  at  4,  since  the  layers  are  there  crowded 
together,  and  conversely  a  rarefaction  exists  in  Figure  154,  d,  at  4, 
since  the  layers  are  there  separated. 

From  the  figures  it  is  evident  that  the  layers,  or  particles,  of 
the  medium  are  in  vibration,  and  that  at  any  instant  these  particles 
possess  a  definite  configuration,  for  example  that  of  particles  4  to 

Z     J      4     S      6       78      9      10     II     12    '3     14-     IS     16 


I     2  3  4     S     6       7 


/        2        3         4        S        6789/0 


/  2  J     4-       S        6          7         8         9      10    //  /2/J 


J  4  S  (>      7      8         9        10       II       /  2     13  14-  /S  /6 


FIG.  154 

16  in  Figure  154,  f.  This  configuration  is  known  as  the  wave  form, 
and  it  travels  from  left  to  right  through  the  medium,  although  the 
particles  themselves  are  merely  vibrating  back  and  forth  across 
their  original  positions.  If  the  motion  of  the  piston  is  simple  har- 
monic, then  the  motion  of  the  particles  must  be  simple  harmonic 
also.  As  a  matter  of  fact  practically  all  vibrations  which  arise 
from  the  elasticity  of  matter  are  of  this  type.  The  amplitude,  or 
maximum  displacement,  of  the  particles  depends  upon  the  ampli- 
tude of  the  motion  of  the  piston,  that  is,  upon  the  intensity  of  the 


224  ELECTRICITY,  SOUND,  AND  LIGHT 

disturbance  which  the  particles  are  propagating.  The  difference 
between  the  times  at  which  any  two  particles  of  the  medium  pass 
through  the  middle  points  of  their  paths,  divided  l>y  the  period  of 
the  vibration,  is  called  the  phase  difference  between  the  particles.  As 
has  already  been  indicated  in  Chapter  XVII,  the  distance  between 
two  successive  particles  which  are  in  similar  states  of  motion  at  the 
same  time  is  called  a  wave  length.  Thus  a  wave  length  is  the  dis- 
tance between  particles  4  and  16,  or  1  and  13,  in  Figure  154,/. 

142.  Transverse  waves.  The  conception  of  wave  motion  just 
given  will  now  be  considered  more  analytically  in  connection  with 
the  transmission  of  transverse  waves.  Thus  if  all  the  particles  in 
the  line  XX'  (Fig.  155,  a)  are  in  some  sort, of  rigid  connection,  and 


ft*. 

•f        -I        •>> . 


FIG.  155 

if  particle  1  is  given  a  displacement  in  a  direction  perpendicular 
to  XX',  then  this  displacement  will  be  successively  communicated 
to  particles  2,  3,  4,  5,  etc.  Further,  if  1  is  made  to  vibrate  with 
simple  harmonic  motion  across  XX1,  then  all  the  particles  2,  3, 
4,  5,  etc.,  will  in  succession  take  up  this  simple  harmonic  motion 
across  XX'',  i.e.  a  series  of  transverse  waves  will  travel  along  XX'. 
Let  the  amplitude  of  the  motion  of  each  particle  be  represented 
by  A  and  the  period  by  T.  Then  it  may  be  shown  that  the  ver- 
tical displacement  yz  of  any  particle,  such  as  2,  expressed  in 
terms  of  A,  T,  and  the  time  t  since  that  particle  left  its  original 
position,  is  as  follows : 

ya=^sm27r-~  (1) 


WAVES  IN   STRINGS 


225 


This  will  be  evident  from  a  consideration  of  Figure  156;  for  a 
particle  P  moving  with  simple  harmonic  motion  along  the  ver- 
tical path  of  length  2  A  has  at  any  instant  a  displacement  from 
the  center  0  which  is  represented  by  the  distance  from  0  to  the 
projection  P  upon  that  path  of  a  point  P'  which  moves  uniformly 
in  a  time  T  about  the  circumference  of  a  circle  which  has  2  A  as 
its  diameter.*  The  angular  speed  of  the  point  P'  about  0  is  2  TT/T 
radians  per  second.  After  t  seconds  the  particle  Pr  has  moved 
through  an  angle  of  2  jrt/T  radians.  The  distance  OP  is  then  A 
times  the  sine  of  this  angle,  i.e. 

y2=^sin27r—  - 

The  displacement  at  this  instant  of  another  particle  such  as  3 
(Fig.  155,/)  is,  of  course,  different.  Thus,  suppose  that  this  par- 
ticle leaves  its  mean  position  t'  seconds  after  particle  2  has  left 
its  mean  position,  i.e.  let  t'/T  be  the  phase  difference  between 
the  two  particles.  The  displacement  of  3  at  the  instant  con- 
sidered is  obviously 


=  A  sin  2  TT 


t-tr 
T 


r 


And  similarly  the  displacement  of  any 
particle  is  represented  by  an  expression 
of  the  form 


y  =  A  sin  2  TT 


t-t' 
T 


2A  - 


FIG.  156 


where  t'/T  is  the  phase  difference  between  the  particle  under  con- 
sideration and  the  reference  particle. 

The  particle  for  which  t'  =  T  will  be  in  a  state  of  motion  similar 
to  that  of  the  reference  particle  2,  and  therefore  distant  from  it  by 
a  wave  length  X.  It  is  also  evident  that  while  the  first  disturb- 
ance which  is  imparted  to  particle  2  is  traveling  forward  this  dis- 
tance X  (for  example,  in  Fig.  loo,/,  from  2  to  14),  the  particle  2 


*See  "Mechanics,  Molecular  Physics,  and  Heat,"  p.  88. 


226  ELECTRICITY,  SOUND,  AND  LIGHT 

makes  one  complete  vibration ;  that  is,  the  wave  travels  X  centi- 
meters in  T  seconds,  or  has  a  velocity  S  which  is  given  by 

S  =  ~  (4) 

Returning  now  to  a  consideration  of  the  motion  of  a  particle 
distant  x  from  particle  2  (for  example,  3,  Fig.  155),  it  is  evident 

x       tr  Tx 

that  -  =  —  •    Substitution  of  the  value  t1  =  —  in  the  expression 
A<       JL  A, 

for  the  vertical  displacement  of  any  particle  gives 

(5) 

The  several  particles  along  the  line  XX'  are  therefore  displaced 
from  their  original  positions  according  to  a  sine  function.  The 
angle  represented  by  2  TTX/\  is  called  the  angle  of  phase  differ- 
ence between  the  particle  under  consideration  and  the  reference 
particle. 

Figure  155,  fr,  c,f,  represents  transverse  displacements  equal  hi 
amount  to  the  longitudinal  displacements  of  Figure  154,  b,  c,  f.  It 
is  also  evident  that  the  confusion  resulting  in  the  figure  for  com- 
pressional  waves  from  the  fact  that  the  displacements  are  parallel 
to  the  direction  of  the  wave  motion  may  be  obviated  by  plotting 
those  displacements  vertically.  With  that  convention  in  mind 
Figure  155,  b,  c,  f,  may  be  taken  to  represent  the  progression 
either  of  a  compressional  or  of  a  transverse  wave.  In  the  discus- 
sion of  nodes  and  loops  in  pipes  (sect.  139,  pp.  215-217)  this  con- 
vention has  already  been  used. 

143.  Stationary  waves.  The  phenomenon  known  as  stationary 
waves  is  the  result  of  the  action  upon  a  series  of  particles  of  two 
equal  trains  of  waves  traveling  in  opposite  directions.  Figure  157,  a, 
shows  two  such  trains  of  waves,  namely  A  traveling  toward  the 
left,  and  B  traveling  toward  the  right.  The  waves  are  represented 
at  an  instant  at  which  the  crests  of  A  are  opposite  to  the  troughs 
of  B,  and  vice  versa.  The  heavy  line  shows  the  resultant  displace- 
ment of  the  series  of  particles.  Obviously  in  this  case  each  one 
of  the  particles  transmitting  the  motion  is  under  the  action  of  two 


WAVES  IN   STRINGS 


227 


disturbances  which  tend  to  produce  equal  and  opposite  displace- 
ments, and  as  a  result  the  particles  suffer  no  displacement  at  all. 
In  Figure  157,  b,  is  shown  the  case  in  which  each  of  the  wave 
trains  has  progressed  an  eighth  of  a  wave  length  ;  that  is,  the  waves 
have  become  displaced  a  quarter  of  a  wave  length  with  respect  to 
one  another.  The  heavy  line  represents  the  form  assumed  by  the 
row  of  particles  at  this  instant  as  a  result  of  the  superposition  of 
the  two  disturbances.  Similarly,  when  one  wave  train  has  moved 
a  half  wave  length  past  the  other  the  resultant  is  again  zero  at 
every  point. 

From  the  figures  it  is  evident  that  for  any  particle  the  resultant 
displacement  is  the  algebraic  sum  of  the  displacements  produced 


FIG.  157 

at  that  point  by  the  two  wave  motions.  For  certain  points,  e.g. 
N,  N'y  N",  distant  from  one  another  by  a  half  wave  length,  the 
resultant  displacement  is  always  zero.  These  points  are  the  nodes, 
and  correspond  to  the  nodes  for  compression al  waves  which  have 
been  previously  discussed.  Between  the  nodes  the  particles  are  in 
constant  vibration,  but  all  pass  through  their  mean  positions  at 
the  same  time.  There  is  therefore  no  phase  difference  between 
successive  particles.  It  is  for  this  reason  that  the  phenomenon  of 
the  combination  of  two  oppositely  directed  trains  of  waves  on  the 
same  particles  is  known  as  the  phenomenon  of  stationary  ivaves. 
The  amplitudes  of  the  vibration  of  successive  particles  vary  from 
a  maximum  at  the  loops  to  zero  at  the  nodes.  On  opposite  sides 


228 


ELECTKICITY,  SOUND,  AND  LIGHT 


of  a  Dode  the  displacements  at  any  instant  are  in  opposite  direc- 
tions. In  Figure  158  are  shown  forms  assumed  by  the  particles 
between  the  two  nodes  drawn  for  successive  instants  of  time.  The 
successive  positions  assumed  by  the  particles  are  numbered  in 
their  order. 


144.  Equation  for  a  stationary  wave.  The  ideas  developed 
above  from  a  study  of  the  diagrams  may  also  be  obtained  from 
a  consideration  of  the  equation  of  a  wave  motion.  For  let 

/  1       x\ 

y  =  A  sin  2  TT  /  -  ---  I  represent  the  equation  for  the  displacement 
\T      X/ 

given  to  the  successive  particles  by  the  direct  wave.  The  reversed 
wave  must  be  one  for  which  at  some  given  instant  of  time  (e.g.  t  =  0) 
the  displacements  given  to  the  same  series  of  particles  will  be  equal 

/  1       x\ 
and  opposite.     The  equation  y1  =  A  sin  2  TT  (  --  1  —  I  satisfies  this 

condition  and  represents  the  equation  of  the  reverse  wave.  The 
resultant  displacement  Y  is  the  sum  of  y  and  yl  .  That  is, 


=  .  sn    ^       - 


sn  + 


By  expansion  *  and  addition  we  have 


TT-  I      C\  A  C\  ^     \  *  C\  V 

Y  =  (  2  A  cos  2  TT  -  I  sm  2  TT  —  • 

A,/  JL 


(6) 


*For 
and 


sin  (0  -f  0)  —  sin  0  cos  0  +  cos  6  sin  0, 
sin  (0  —  0)  =  sin  0  cos  0  —  cos  6  sin  0. 


WAVES  IN   STBINGS 


229 


Now   consider  the  fundamental  equation  for  a  wave  motion, 

(t       x\ 
—  —  -  I  >  and  notice  that  it  consists  of  two 
1       A./ 

parts.    The  first  part  is  the  amplitude  A  and  the  second  a  sine  func- 
tion of  the  time  t  plus  a  phase  constant  x/\.    Now  in  the  expression 

for  Y  just  obtained  sin27r —  is  the  sine  function 

of  the  time.  And  evidently,  since  there  is  no 
phase  constant  in  this  expression,  the  particles 
must  all  be  in  the  same  phase  of  vibration.  Simi- 
larly 2  A  cos  2  TT  -  represents  the  amplitude.  But 
\ 

since  x  represents  the  distance  of  the  particle 
under  consideration  from  the  reference  particle,  it 
is  evident  that  the  amplitude  varies  for  successive 

/v» 

particles.    Also  since  cos  2  TT  —  is  zero  when  -x  is  an 

\ 

odd  multiple  of  X/4,  it  follows  that  there  are  nodes, 
or  points  of  zero  amplitude,  at  points  differing  succes- 
sively by  half  wave  lengths.  Further,  the  algebraic 

rvi 

sign  of  cos  2  TT  —  changes  at  these  same  points. 

A, 

145.  Melde's  experiment.  A  capital  illustration  of 
stationary  waves  in  strings  is  furnished  by  what  is 
commonly  known  as  Meldes  experiment.  One  end 
of  a  light  cord  is  attached  to  one  of  the  prongs  B 
(Fig.  159)  of  a  tuning  fork,  while  the  other  end 
carries  a  weight  W. 

The  waves  which   start   down   the  cord  from  the 
vibrating  fork  are  reflected  at  W,  so  that  two  trains      FIG.  159 
of  waves  moving  in  opposite  directions  become  super- 
posed upon  the  cord.    In  accordance  with  the  principles  of  the 
last  section  this  condition  tends  to  give  rise  to  stationary  waves, 
the  positions  of  the  nodes  being  at  distances  from  W  correspond- 
ing to  exact  multiples  of  a  half  wave  length  of  the  train  sent 
down  the  cord  from  the  fork.    Since,  however,  the  upward-moving 
train  is  again  reflected  at  B,  the  condition  for  stationary  waves  in 
which  the  nodes  are  at  distances  from  B  corresponding  to  exact 


230  ELECTRICITY,  SOUND,  AND  LIGHT 

multiples  of  a  half  wave  length  is  also  established.  It  is  obvious 
that  both  of  these  conditions  can  be  met,  and  permanent  stationary 
waves  set  up  in  the  string,  only  if  the  length  L  of  the  string  is  an 
exact  multiple  of  a  half  wave  length.* 

Instead  of  varying  the  length  of  the  string  so  as  to  fulfill  this 
condition*,  it  is  customary  to  vary  the  wave  length  by  varying  the 
load  W.  For  the  wave  length  X  is  connected  with  the  vibration 
rate  n  of  the  fork  and  the  speed  of  propagation  S  of  the  train  of 
waves  along  the  string  by  means  of  the  relation  S  =  n\  and  the 
speed  S  is  connected  with  the  tension  T  in  the  string  and  its 
mass  p  per  centimeter  of  .length  by  means  of  a  formula  which  is 
very  similar  to  that  given  in  the  last  equation  on  page  189.  It  is, 


*  This  statement  is  only  approximately  correct,  since  the  end  of  the  fork  is 
not  exactly  at  a  node,  but  rather  just  as  near  to  a  node  as  a  point  near  some 
other  node  which  has  the  same  amplitude  of  vibration  as  the  fork. 

t  This  formula  is  most  satisfactorily  deduced  with  the  aid  of  the  calculus, 
but  it  may  also  be  obtained  as  follows.  Let  the  curve  mno  (Fig.  160)  represent 
a  portion  of  the  cord  over  which  the  deformation  is  being  propagated.  Let  ee' 
be  an  element  of  the  cord  so  small  that  it  may  be  considered  as  the  arc  of  a 
circle  of  radius  R.  If  the  string  is  wholly  devoid  of  rigidity,  then  the  only  force 
which  is  urging  the  element  toward  the  center  c  arises  from  the  tension  T  in  the 
string,  and  this  may  be  regarded  as  a  pull  acting  upon  each  end  of  the  arc  ee'. 


FIG.  160 


The  directions  of  these  two  pulls  are  the  directions  of  the  tangents  to  the  curve 

at  the  points  e  and  e'  respectively  (see  Fig.  160).    The  component  of  each  of 

i  gg/ 
these  pulls  which  is  urging  ee'  toward  c  is  T  sin  6  =  T  2 The  total  force  /c 

C6*  CO* 

which  is  urging  ee'  toward  c  is  therefore  2  T =  T  — •    But  since  the  defor- 

2  .R  R 

mation  mno  is  propagating  itself  unchanged  in  character  along  the  string,  each 
element  of  the  string  must  assume  in  succession  the  positions  occupied  at  any 
instant  by  all  the  other  elements  of  the  curve  mno.  In  other  words,  at  the  instant 
which  we  have  been  considering  the  elem'/nt  ee'  is  not  moving  at  all  in  the 


WAVES  IK   STRINGS  231 

so  that  by  varying  the  tension  T  it  should  be  possible  to  find  a 
whole  series  of  values  of  X  which  will  give  rise  to  permanent 
stationary  waves  in  the  cord.  Thus  when  L  =  X/2  the  string  should 
vibrate  in  1  segment,  when  L  =  2  X/2  it  should  vibrate  in  2  seg- 
ments, when  L  =  3  X/2,  in  3  segments,  etc. 

Or,  in  general,  since  by  combining  equation  (7)  with  the  equation 
we  obtain 

(10) 


it  is  evident  that  the  equations  of  condition  for  1,  2,  3,  4,  etc., 
segments  may  be  written 

1       (4T 
P 

These  equations  are  obtained  by  substituting  in  (10)  the  above 
relations  between  L  and  X/2.  Equations  (11)  show,  since  n,  L, 
and  p  do  not  change,  that  the  product  of  the  stretching  force  by  the 
square  of  the  number  of  segments  should  be  a  constant,  and  that 
this  constant  should  represent  the  tension  when  the  entire  string 
is  vibrating  in  one  segment.  By  substituting  this  constant  in  the 
first  of  equations  (11),  T  having  been  expressed,  of  course,  in  dynes, 
it  should  be  possible  to  find  the  vibration  rate  n  of  the  fork,  pro- 
vided p  and  L  are  known.  The  fact  that  all  of  these  relations 
are  found  by  experiment  to  hold,  constitutes  complete  experimental 

\T 
proof  of  the  correctness  of  the  formula  S=  \l — >  for  the  case  of 

strings  which  have  no  rigidity. 

direction  of  c,  but  is  instead  moving  into  the  position  of  the  adjacent  element 
on  mno;  that  is,  it  is  moving  with  a  velocity  S  along  the  circumference  of  the 
circle  which  has  R  for  its  radius.  Hence  we  may  apply  to  its  motion  the  law 
of  centripetal  force  deduced  on  page  102  of  "  Mechanics,  Molecular  Physics, 
and  Heat,"  namely,  »,02 

fc=~  (8) 

But  if  p  is  the  mass  per  unit  length  of  the  string,  m  =  ee'p. 

Hence  (8)  becomes  fr  =  — 

R 

But  since  fc  is  also  equal  to  Tee'/R,  we  have 

— - —  = ,       or       S  =  \  \  —  •  (9) 

R  R  \P 


232  ELECTRICITY,  SOUND,  AND  LIGHT 

146.  Fundamentals  and  overtones  in  strings.  If  a  stretched 
string  is  plucked  in  the  middle,  the  deformation  travels  in  opposite 
directions  to  the  two  ends,  is  there  reflected,  and,  since  the  two 
reflected  portions  returning  to  the  middle  unite  in  like  phases  at 
this  point,  the  net  result  of  the  propagation  of  the  disturbance 
back  and  forth  over  the  string  is  a  vibration  of  the  string  as  a 
whole  in  the  manner  indicated  in  Figure  158,  page  228,  in  which 
the  various  lines  represent  some  of  the  successive  positions  of  the 
string.  A  string  vibrating  in  this  way  imparts  successive  con- 
densations and  rarefactions  to  the  air  in  which  it  moves,  and 
these,  being  transmitted  to  the  ear,  give  rise  to  a  note  of  a  definite 
pitch  which  is  called  the  fundamental  note  of  the  string.*  Since 
the  time  elapsing  between  the  instant  at  which  the  string  is  in  the 
position  AcB  (Fig.  158)  and  the  instant  at  which  it  assumes  the 
position  AclB  is  the  time  required  for  the  deformation  to  travel 
over  the  paths  cBd  and  cAd,  it  will  be  seen  that  during  the  time 
of  one  half  vibration  of  the  string  the  disturbance  travels  on  the 
string  a  distance  exactly  equal  to  the  length  of  the  string.  Hence 
during  the  period  of  one  complete  vibration  of  the  string  the  dis- 
turbance travels  twice  the  length  of  the  string.  Thus  we  arrive, 
from  a  wholly  different  point  of  view,  at  the  conclusion  of  the  pre- 
ceding section,  namely,  that  when  a  string  is  vibrating  as  a  whole, 
Le.  in  one  segment,  its  length  is  one  half  the  wave  length  of  the 
waves  which  are  traveling  back  and  forth  over  it. 

If  the  string  is  clamped  in  the  middle  as  wTell  as  at  the  ends 
and  plucked  one  fourth  of  its  length  from  one  end,  each  half  vibrates 
precisely  as  the  whole  string  vibrated  in  the  preceding  case ;  but 
since  the  speed  of  propagation  is  the  same  as  before,  while  the  dis- 
tance between  reflections  is  one  half  as  great,  the  period  of  vibration 
of  each  half  of  the  string  must  be  one  half  as  great  as  the  preced- 
ing period.  Hence  the  note  communicated  to  the  air  is  the  octave 
of  the  original  note,  and  the  wave  length  of  the  note  is  the  length 
of  the  string.  The  note  thus  produced  by  the  string  is  called  its 

*  Practically,  of  course,  the  sound  thus  derived  is  of  small  intensity,  and  in 
most  musical  instruments  the  greater  magnitude  of  sound  is  due  to  synchronous 
vibrations  which  the  string  impresses  upon  its  supports  and  through  them  upon 
sounding-boards  and  resonant  volumes  of  air. 


WAVES  IN   STRINGS  233 

first  overtone.    If  the  string  is  not  clamped  in  the  middle,  but 
is  plucked  one  fourth  of  its  length  from  one  end,  it  still  tends 
to  vibrate  as  above  in  two  segments,  but  this  vibration  is  super- 
posed upon  the  vibration  of 
the  string  as  a  whole,  so  that 
the  fundamental  and  the  first 
overtone  can  be  heard  simul- 
taneously.   Figure  161  is  an  FlG  161 
endeavor  to  show  the  appear- 
ance of  a  string  which  is  vibrating  so  as  to  produce  its  funda- 
mental and  first  overtone. 

Similarly,  if  the  string  is  plucked  one  sixth  of  its  length  from 
one  end,  it  tends  to  vibrate  in  three  segments  and  the  second  over- 
tone will  be  heard  with  the  fundamental. 

Thus  the  string  is  capable,  under  suitable  conditions,  of  vibrat- 
ing in  any  number  of  segments  and  of  giving  out  a  series  of  notes 
whose  frequencies  bear  to  the  fundamental  frequency  the  ratios 
2,  3,  4,  5,  6,  7,  etc.  In  general,  in  the  case  of  the  strings  of  musical 
instruments  several  of  these  overtones  are  produced  simultaneously 
with  the  fundamental,  which  ones  are  present  depending  chiefly 

upon  where  the  string  is  struck 

"-•-"----'.'JJ.."',    or  bowed.     It  is  to  differences 

in  the  number  or  relative  promi- 
nence of  the  overtones  that  all 

---"" """"'^     >•'''  differences   in  the   qualities   of 

~"~— - -'"     %^-.  . .    different  notes  of  the  same  pitch 

are  assigned. 

^ .^  147.  Transverse  waves  in 

c      ^,--:le:-.._^  ,_-"'':*:^      rods.    In  the  case  of  rods  the 

wave  travels  as   the  result  of 
the  rigidity  of  the  substance  of 

a      "":y --y-:-    '.'-"y^""    which  the  rod  is  composed.  'A 

FIG  1G2  consideration  of  the  velocity  of 

propagation  of  the  wave  is,  how- 
ever, beyond  the  scope  of  this  text.  The  form  assumed  by  vibrat- 
ing bars  may  be  seen  from  Figure  162.  A  bar  clamped  at  one 
end  gives  off  its  fundamental  note  when  vibrating  in  the  form 


234  ELECTRICITY,  SOUND,  AND  LIGHT 

shown  in  Figure  162,  a.  If  struck  more  sharply  and  nearer  the 
free  end,  it  may  be  made  to  give  off  its  first  overtone ;  in  this 
case  it  vibrates  in  the  form  shown  in  Figure  162,&.  The  relation 
between  the  frequencies  of  the  fundamental  and  its  various  over- 
tones are  not,  however,  simple  numbers,  as  is  the  case  with  pipes 
or  strings. 

If  the  rod  is  supported  at  two  points,  as  in  Figure  162,  c,  it  will 
vibrate  in  the  form  shown  in  that  figure  when  yielding  its  funda- 
mental note.  The  form  assumed  by  the  rod  when  yielding  its  first 
overtone  is  shown  in  Figure  162,c?.  In  this  case  also  the  relation 
of  the  frequencies  is  not  a  simple  number. 

148.  The  tuning  fork.    If  the  rod  shown  in  Figure  162,  c,  is 
bent,  it  is  found  that  the  nodes  are  brought  closer  together.    If  it 

has  the  form  of  Figure  163,  the  nodes  will  occur  at 
the  points  marked  NN.  The  higher  overtones  are 
then  very  difficult  of  production  and  are  very  much 
less  in  intensity  than  the  fundamental.  A  bar  bent 
into  this  form  and  supported  at  P  is  known  as  a  tun- 
ing fork.  Because  of  the  purity  of  its  note  —  that  is, 
the  absence  of  overtones  —  it  has  been  adopted  for 
use  as  a  convenient  standard  of  frequency.  Any  given 
fork  must,  of  course,  be  rated  first  by  some  absolute 
method  and  then  it  may  be  used  for  comparison  with 
other  sources  of  sound. 

149.  Beats.    The  case  of  stationary  waves  (see  sects.  143-144) 
is  a  special  case  of  interference.    Another  case  of  especial  interest 
in  sound  is  the  interference  of  the  wave  trains  from  two  musical 
sources  of  sound  of  almost  the  same  pitch  (Le.  frequency).   Obviously 
if  a  crest  due  to  one  source  reaches  the  ear  at  the  same  time  as  the 
trough  from  the  other  source,  there  will  be  destructive  interference. 
But  since  one  source  is  vibrating  slightly  more  rapidly  than  the 
other,  an  instant  later  two  crests  (or  two  troughs)  will  be  in  coin- 
cidence at  the  ear.    There  results   then  a  reenforcement  of  the 
vibration.    These  alternations  in  the  intensity  of  the  sound  at  any 
point  will  obviously  occur  as  many  times  per  second  as  the  fre- 
quency of  one  source  exceeds  that  of  the  other.    That  is,  if  m  and 
n  represent  the  frequencies  of  the  two  sources,  there  will  be  m-n 


WAVES  IN   STRINGS  235 

such  coincidences  per  second.  These  alternations  in  sound  intensity 
are  known  as  beats. 

The  phenomenon  of  beats  is  useful  in  explaining  the  physical 
basis  of  discords.  So  long  as  the  number  of  beats  produced  by 
sounding  two  notes  together  is  not  more  than  five  or  six  per 
second  the  effect  is  not  particularly  unpleasant.  From  this  point 
on,  however,  the  beats  begin  to  become  indistinguishable  as  sepa- 
rate beats  and  pass  over  into  a  discord.  The  unpleasantness  be- 
comes worst  at  a  difference  of  about  thirty  vibrations  per  second. 
Thus  the  notes  B  and  C1,  which  differ  by  thirty-two  vibrations 
per  second,  produce  about  the  worst  possible  discord.  When  the 
difference  reaches  as  much  as  seventy,  the  difference  between  C 
and  E,  the  effect  is  again  pleasing  or  harmonious. 

But  in  order  that  two  notes  may  harmonize  it  is  necessary  not 
only  that  they  themselves  should  not  produce  an  unpleasant  num- 
ber of  beats,  but  that  their  overtones  also  should  not  do  so.  Thus 
C  and  B  are  very  discordant  although  they  differ  by  a  large  num- 
ber of  vibrations  per  second.  •  The  discord  arises  in  this  case  between 
B  (vibration  number  480)  and  C',  the  first  overtone  of  C  (vibration 
number  512).  Thus  if  two  notes  are  to  be  consonant,  neither  they 
nor  any  of  their  overtones  can  fall  close  enough  together  to  produce 
an  unpleasant  number  of  beats. 

EXPERIMENT  20 

V^T 
— ,  and  to  determine  the  rate 

of  a  given  tuning  fork. 

Directions.  Set  up  an  electrically  driven  fork  having  from  about  100  to 
300  vibrations  per  second  and  connect  as  shown  in  Figure  159.  Connect  to 
the  binding  posts  of  the  fork  a  single  storage  cell.*  To  one  prong  of  the 
fork  attach  a  light  string,  for  example  a  piece  of  oiled  fish  line  or  linen 
thread  about  four  feet  in  length.  To  the  other  end  of  this  string  hang  a 
light  pan  for  holding  weights.  Xow  vary  the  tension  by  adding  weights 
until  the  string  breaks  up  into  some  number  of  vibrating  segments.  The 
adjustment  may  be  made  exact  by  varying  the  tension  until  the  nodes  are 

*  If  it  is  necessary  to  supply  more  energy  to  the  fork  in  order  that  it  may 
not  be  damped  down  by  the  tension  on  the  string,  a  larger  number  of  cells  may 
be  used.  To  prevent  excessive  sparking  at  the  break  a  condenser  may  be  shunted 
across  it,  as  is  clone  in  the  induction  coil. 


236  ELECTRICITY,  SOUND,  AND  LIGHT 

most  sharply  defined.  It  is  desirable  to  use  values  of  the  tension  for  which 
there  will  result  not  more  than  seven  half  waves.  If  the  tension  necessary 
to  produce  this  result  is  sufficient  to  stop  the  vibrations  of  the  fork,  follow 
the  instructions  of  the  footnote  on  the  preceding  page,  or  shorten  the  length 
of  string  used. 

Vary  the  tension  and  note  the  weights  corresponding  to  at  least  three 
different  wave  lengths  such  that  the  numbers  of  half  wave  lengths  in  the 
length  of  the  string  are  successive  numbers ;  for  example,  3,  4,  5.  See  how 
nearly  a  constant  number  you  obtain  by  multiplying  the  tension  by  the 
square  of  the  number  of  segments  (see  sect.  145). 

Weigh  the  string  and  measure  its  length.  Calculate  its  linear  density  p. 
Express  in  dynes  the  tension  necessary  to  cause  the  string  to  vibrate  in  one 
segment.  Calculate  from  these  data  the  number  of  vibrations  per  second 
made  by  the  fork  (see  eq.  (11)). 

(B)  Object.  To  find  the  vibration  frequency  of  the  fork  used  in  (A)  by 
the  method  of  beats. 

Directions.  It  was  the  conclusion  of  the  discussion  of  section  149  that 
if  two  sources  of  sound  differ  slightly  in  their  vibration  frequencies,  there 
results  in  a  second  of  time  a  number  of  alternations  in  the  intensity  of  the 
resultant  sound  that  is  equal  to  the  difference  between  the  frequencies  of 
the  two  sources.  If  one  of  the  sources  is. of  known  frequency, — for  exam- 
ple, a  standard  fork, — the  frequency  of  the  second  source  may  be  deter- 
mined by  observing  the  number  of  these  "  beats."  The  unknown  fork  to 
be  used  is  that  of  the  preceding  part  of  this  experiment.* 

Select  by  ear  a  standard  fork  of  about  the  same  note  as  the  unknown. 
Always  set  the  fork  in  vibration  by  striking  it  with  a  felt-covered  hammer 
like  a  piano  hammer,  or  with  a  rubber  mallet.  The  latter  is  conveniently 
made  by  placing  a  rubber  stopper  on  the  end  of  a  rod.  Now,  using  a  stop 
watch,  count  the  number  of  beats  for  several  seconds.  If  the  number  is 
large,  it  will  be  found  easier  to  count  the  beats  in  groups  of  three  or  four, 
e.g.  one,  two,  three ;  one,  two,  three,  etc.  Now  make  the  frequency  of  the 
unknown  less  by  attaching  to  one  of  its  prongs  a  small  piece  of  soft  wax. 
Count  again  the  number  of  beats.  If  this  number  is  less  than  before, 
obviously  the  unknown  has  been  brought  nearer  the  standard  by  weighting ; 
that  is,  its  vibration  frequency  is  larger  than  the  standard  by  the  number 
of  beats  first  observed.  If  the  act  of  weighting  the  unknown  fork  increases 
the  number  of  beats,  then  the  number  of  vibrations  per  second  of  the 
unknown  i^f  smaller  than  that  of  the  known.  Hence  the  number  of  beats 
per  second  must  be  added  to  the  vibration  number  of  the  standard  to  obtain 
that  of  the  unknown  fork. 


*  Or  it  may  be  the  fork  of  the  falling-body  apparatus,  or  of  the  inertia  disk 
of  Experiments  1  or  10  of  "Mechanics,  Molecular  Physics,  and  Heat."  Either 
of  these  should  be  rated  in  the  supporting  frame  and  not  removed. 


WAVES  IX   STRINGS  237 

EXAMPLE 

(A)  The  tension  for  3  segments  was  162  g.  ;  for  4  segments,  94  g. ;  for 
5  segments,  60  g. ;  for  6  segments,  42  g. ;  for  7  segments,  30  g.   The  products 
of  the  tension  in  grams  and  the  square  of  the  number  of  segments  were  1458, 
1504,  1500,  1512,  and  1470  respectively.    The  average  product  was  1488. 
The  length  of  the  string  was  95.4  cm.;  its  mass,  .390  g.;  its  linear  density, 
.00409.    Multiplying  1488  by  980  to  reduce  the  tension  to  dynes  and  then 
substituting  the  above  values  of  T,  p,  and  L  in  the  first  of  equations  (11), 
gave  n  =  99.    The  fork  used  was  marked  by  the  maker  100  vibrations  per 
second. 

(B)  Using  the  fork  and  a  standard  of  frequency  100,  one  beat  in  two 
seconds  was  observed.    Weighting  the  unknown  increased  the  number  of 
beats;  hence  the  rate  of  the  unknown  was  99.5  per  second  and  the  per  cent 
of  error  in  (A)  was  .5. 


CHAPTEE  XXI 
DIFFRACTION  OF  SOUND  AND  LIGHT  WAVES 

150.  Two  theories  of  light.  In  Sir  Isaac  Newton's  day 
(1642-1727)  two  rival  theories  of  light  were  struggling  for  recog- 
nition. The  one,  the  wave  theory,  fathered  and  championed  by 
the  Dutch  physicist,  Christian  Huygeiis  (1629-1695),  regarded 
light,  like  sound,  as  some  sort  of  a  wave  motion,  the  chief  differ- 
ence between  the  two  being,  according  to  this  theory,  that,  while 
sound  is  propagated  through  the  agency  of  ordinary  matter,  light 
is  a  wave  motion  in  some  all-pervading  medium  to  which  the  name 
of  "  the  ether  "  was  given. 

The  rival  theory,  called  the  corpuscular  theory,  regarded  light 
as  due  to  the  emission  from  all  luminous  bodies  of  minute  cor- 
puscles which  travel  in  straight  lines  and  with  enormous  velocities 
through  space  and  produce  the  sensation  of  light  when  they  impinge 
upon  the  retina  of  the  eye.  This  theory  had  its  most  famous  and 
most  brilliant  advocate  in  Sir  Isaac  Newton  himself. 

Newton's  chief  reason  for  rejecting  the  wave  theory  lay  in  the 
fact  that  he  was  unable  to  understand  why,  if  light  is  a  wave 
motion,  it  is  always  propagated  in  straight  lines  past  the  edges  of 
opaque  objects,  instead  of  undergoing  diffraction,  that  is,  being  bent 
around  such  objects,  as  are  sound  waves,  water  waves,  and  all  the 
other  types  of  waves  with  which  Newton  was  familiar.  What  is  com- 
monly regarded  as  the  decisive  test  between  the  two  theories  was 
made  in  the  year  1800  by  Thomas  Young,  and  consisted  in  show- 
ing that  it  is  possible  to  produce  with  light  waves  the  diffraction 
phenomenal  which  are  to  be  discussed  in  the  later  sections  of  this 
chapter,  and  which  it  does  not  seem  possible  to  account  for  from 
the  standpoint  of  the  corpuscular  theory. 

It  is  the  object  of  the  present  discussion  and  of  the  succeeding 
experiments  to  show  both  theoretically  and  experimentally  that, 
under  suitable  conditions,  sound  does  not  bend  around  corners,  as 

238 


DIFFKACTION  OF  SOUND  AND  LIGHT  WAVES    239 

it  is  commonly  supposed  invariably  to  do,  and  that  light,  on  the 
other  hand,  does  under  suitable  conditions  bend  around  corners, 
as  it  is  commonly  supposed  not  to  do.  More  explicitly  stated,  our 
aim  will  be  to  show  that  the  phenomenon  of  straight-line  propa- 
gation is  characteristic  of  any  and  all  types  of  wave  motion,  pro- 
vided only  the  aperture  through  which  the  waves  pass  is  large  in 
comparison  with  the  wave  length  of  the  waves.  If  this  proposition 
can  be  proved,  it  will  be  evident  that  the  fact  of  the  straight-line 
propagation  of  light  does  not  furnish  any  argument  against  the 
wave  theory,  provided  the  wave  length  of  ordinary  light  waves  is 
very  minute  in  comparison  with  the  dimensions  of  ordinary  aper- 
tures. Before  proceeding  to  this  proposition  it  is  necessary  to  con- 
sider further  the  nature  of  a  wave  motion  in  a  medium  of  indefinite 
extent,  and  the  conditions  for  interference  in  such  a  medium. 

151.  Definition  of  wave  front.  Consider  S  in  Figure  164  to 
be  the  point  source  of  a  \vave  motion  in  an  isotropic  medium ; 
that  is,  a  medium  in  which  the  disturbance  is  propagated  with 
equal  speed  in  all  directions.  When  the 
disturbance  which  originates  at  S  has  just 
reached  a,  it  has  also  then  just  reached  all 
other  points,  such  as  b,  c,  and  d,  which  are 
at  the  same  distance  from  S.  The  spher-  d\ 
ical  surface  passing  through  these  points  is 
known  as  the  wave  front  of  the  disturbance. 
In  general,  the  wave  front  may  be  defined 

as  the  surface  passing  through  all  the  particles  which  are  in  the 
same  phase  of  vibration. 

The  form  of  the  wave  front  under  the  conditions  just  mentioned 
is  spherical,  but  it  will  be  shown  later  that  conditions  may  arise 
in  which  it  has  not  this  form.  Further,  it  will  also  be  shown  that 
under  proper  conditions  a  spherical  wave  may  be  converging,  i.e. 
concave  toward  the  direction  in  which  it  is  traveling,  instead  of 
diverging,  as  in  the  case  just  considered.* 

If  the  source  is  far  enough  away  (rigorously,  at  an  infinite 
distance),  the  wave  front  will  obviously  be  plane. 

*  See  section  155,  page  245. 


240  ELECTRICITY,  SOUND,  AND  LIGHT 

152.  Construction  of  a  wave  front.   Any  particle  in  the  wave 
front  of  a  disturbance  may  be  considered  as  a  point  source  from 
which  is  spreading  out  a  spherical  wave.    Thus  consider  the  par- 
ticles «,  b,  c,  and  d  in  the  plane  wave  represented  in  Figure  165. 
A  short  time  after  the  disturbance  has  reached  these  particles  let 
the  spherical  wave  surfaces  due  to  them  have  the  forms  shown  in 
the  figure.     If  the  number  of  these  new  centers  is  very  large,  it  is 
evident  from  the  figure,  where  for  clearness  only  four  have  been 
represented,  that  the  disturbance  along  the  surface  AD  is  very 
much  greater  than  at  any  other  points.    In  fact  it  may  be  shown 
by    a    mathematical   analysis    that  these   small   spherical  waves 
destroy   each   other    by   interference   except   at   the   surface  AD. 

From  its  geometrical  nature  as 
a  surface  tangent  to  all  the 
smaller  surfaces,  AD  is  known 
as  their  envelope.  This  envelope 
is  then  the  new  wave  front  of 
j,  ,  1C5  the  disturbance.  In  general  it 

may  be  said  that  at  any  instant 

the  wave  front  of  a  disturbance  is  the  envelope  of  all  the  secondary- 
wave  surfaces  which  are  due  to  tlie  action  as  separate  sources  of  all 
the  various  particles  that  at  some  previous  instant  constituted  the 
wave  front. 

153.  Conditions  for  interference  of  two  wave  trains  in  a  medium 
of  indefinite  extent.   Let  A  and  B  (Fig.  166)  be  two  particles  vibrat- 
ing in  the  same  phase  from  each  of  which  is  propagated  a  disturb- 
ance having  a  spherical  wave  front.    Let  similar-  wave  fronts  be 
constructed  for  each  particle.     Thus  the  circular  arcs  a  and  a' 
represent  the  position  of  particles  at  the  same  distance  from  their 
respective  sources  and  therefore  in  the  same  phase.    The  arcs  I  and 
y  represent  the  wave  fronts  when  the  disturbances  have  traveled 
one  half  a  wave  length  farther;  that  is,  each  of  them  represents 
the  locus  of  a  series  of  particles  which  are  exactly  opposite  in 
the  phase  of  their  vibration  to  the  particles  of  a  and  af.    The  arcs 
c  and  cf  represent  the  wave  fronts  when  they  have  traveled  a 
whole  wave  length  beyond  a  and  a1.    Their  particles  are  in  similar 
phase  to  those  of  a  and  a'  and  opposite  to  those  of  b  and  I'. 


DIFFBACTION  OF  SOUND  AND  LIGHT  WAVES     241 


The  particles  in  the  line  determined  by  the  points  marked  x, 
xz,  xz  have  superimposed  upon  them  vibratory  motions  of  the 
same  phase  from  both  sources.  Along  this  line  there  is  therefore 
a  reenforcement,  or  a  maximum  disturbance.  Along  the  line  deter- 
mined by  the  points  marked  ov  02,  03,  on  the  other  hand,  the  vibra- 
tions superimposed  are  opposite  in  phase,  and  there  is  interference, 
or  a  minimum  disturbance.  Further,  along  the  line  determined 


by  the  points  xv 


there  is  again  reenforcement.    From  the 


construction  of  the  figure 
it  is  evident  that  the  con- 
dition for  a  maximum  at 
any  point  is  the  existence 
of  a  difference  in  length 
of  path  between  the  point 
and  the  sources  A  and  B 
respectively  of  some  in- 
tegral multiple  of  a  whole 
wave  length.  Thus  at  x 
the  difference  in  path  is 
zero  wave  lengths,  at  xl  it 
is  one  wave  length,  etc. 
Similarly,  for  a  mini- 
mum the  difference  in 
distance  must  be  an  odd 
multiple  of  a  half  wave 
length.  At  ov  o2,  og,  etc., 
it  is  ^  wave  length.  Ad- 
ditional maxima  and 
minima  may  be  found  by  extending  the  lines  a, 

It  is  important  to  notice  that  the  lines  of  minimum  disturbance 
ov  02,  03,  etc.,  move  farther  and  farther  away  from  the  central 
line  of  maximum  disturbance  x,  xz,  xz,  the  smaller  the  distance 
AS  becomes  in  comparison  with  a  wave  length.  Thus  if  AB  is 
very  large  in  comparison  with  a  wave  length,  the  line  ov  o2,  os 
is  very  close  to  the  line  x,  xz,  xs,  and  similarly  the  line  xv  #4,  x5 
is  close  to  the  line  ov  ofl,  os.  But  as  AB  becomes  smaller  and 
smaller  these  lines  diverge  more  and  more.  When  AB  is  just 


FIG.  166 


6,  V,  etc. 


242  ELECTRICITY,  SOUND,  AND  LIGHT 

equal  to  a  wave  length  the  line  xv  #4,  x&  is  in  the  prolongation 
of  AB,  since  it  is  only  points  in  this  line  which  can  then  differ 
by  one  wave  length  in  their  distances  from  A  and  B  respectively. 
When  AB  is  equal  to  a  half  wave  length  the  line  ov  o2,  o3  is  in 
the  prolongation  of  AB,  and  there  are  then  no  points  of  quiescence 
at  all  to  the  right  of  AB.  When  AB  is  less  than  a  half  wave 
length  there  are  no  points  of  quiescence  anywhere.  These  con- 
siderations will  now  be  applied  to  the  discussion  of  the  rectilinear 
propagation  of  wave  disturbances  through  openings  in  screens. 

154.  The  propagation  of  wave   motions   through   apertures. 
Consider  the  case  of  a  train  of  short  waves,  which,  proceeding  from 
a  distant  source,  pass  through  an  opening  ac  (Fig.  167)  and  fall 
upon  a  screen  mn.    Assume  that  the  length  ac  is 
large  as  compared  with  the  wave  length  of  the  train 
of  waves.    A  distant  source  is  chosen  so  that  the 


wave  front  of  the  disturbance  which  reaches  the 


FIG.  167 

aperture  ac  may  be  practically  a  plane,  and  thus 
admit  of  the  consideration  of  all  the  particles  lying 
in  the  plane  of  the  aperture  as  being  in  the  same 
phase  of  vibration. 

The  lines  ao  and  cr  are  drawn  from  the  source,  assumed  to  be  a 
point,  past  the  edges  of  the  opening  ac  to  the  screen ;  i.e.  they  are 
the  lines  which  mark  the  limits  of  the  geometrical  learn.  Suppose 
that  the  wave  length  and  the  opening  ac  are  so  related  that  the 
point  p2  on  the  screen,  for  which  the  distance  cp2  is  exactly  one 
wave  length  greater  than  the  distance  apz,  falls  outside  the  limits 
of  the  geometrical  beam,  i.e.  above  the  point  o.  Then  the  particles  a 
and  b  will  differ  in  distance  to  p2  by  a  half  wave  length.  Hence  the 
vibrations  produced  at^>2  by  these  two  particles  mutually  neutralize 


DIFFBACTION  OF  SOUND  AND  LIGHT  WAVES     243 

each  other.  Similarly  the  disturbance  originating  in  the  first  par- 
ticle below  a  will  at  p2  be  just  one  half  wave  length  ahead  of  the 
disturbance  coming  from  the  first  particle  below  b.  Thus  every 
particle  between  a  and  b  may  be  paired  off  with  a  corresponding 
particle  between  b  and  c  such  that  the  effects  of  the  two  particles 
neutralize  each  other  at  p2.  Hence  the  total  effect  at  p2  of  the 
disturbances  coming  from  the  portion  ab  of  the  opening  is  com- 
pletely neutralized  by  the  effect  of  the  disturbances  coming  from 
the  portion  be  of  the  opening. 

Consider  next  a  point  p4  which  is  so  situated  that  the  distance 
cpt  is  two  wave  lengths  more  than  the  distance  ap4.  The  opening 
ac  may  now  be  divided  into  four  parts,  ae,  eb,  bf,fc,  such  that  eb 
neutralizes  at  p±  the  effect  of  -ae,  since  ep±  is  one  half  wave  length 
more  than  ap±,  and  fe  neutralizes  the  effect  of  bf,  since  fp±  is  one 
half  wave  length  more  than  bp±.  There  is  therefore  no  disturbance 
at  all  at  p±. 

At  some  point  p3,  between  p2  and  p4,  the  distance  cps  will  be  one 
and  a  half  wave  lengths  more  than  aps.  If  we  now  divide  ac  into 
three  equal  parts,  the  effect  of  the  upper  third  will  be  completely 
neutralized  at  ps  by  that  of  the  next  lower  third,  but  the  effect  of 
the  lowest  third  has  nothing  to  neutralize  it  at  p3 ;  hence  there  is  a 
disturbance  at  pz  which  is  due  simply  to  one  third  of  the  particles 
between  a  and  c,  and  even  the  effects  of  the  particles  in  this  third 
partially  neutralize  one  another  at  ps,  since  they  differ  somewhat 
in  phase.  It  is  obvious  that  between  p%  and  p^  the  disturbance 
increases  from  zero  at  p2  to  a  maximum  at  p3,  and  then  falls  grad- 
ually to  zero  at  p± ;  that,  further,  there  are  other  points  of  zero 
disturbance,  pQ,  etc.,  so  situated  that  the  distance  from  c  to  the 
point  in  question  is  any  even  number  of  half  wave  lengths  more 
than  the  distance  from  a  to  this  point;  and  that  between  these 
points  of  zero  disturbance  are  points  of  maximum  disturbance,  p5, 
etc.,  so  situated  that  the  distance  from  c  to  the  point  in  question  is 
any  odd  number  of  half  wave  lengths  more  than  the  distance  from 
a  to  this  point.  But  it  will  also  be  noticed  that  the  successive 
maxima,  p%,  p5,  etc.,  diminish  rapidly  in  intensity,  since,  while  but 
two  thirds  of  the  particles  between  a  and  c  completely  neutralize 
one  another's  effects  at  p%,  four  fifths  of  these  particles  neutralize 


244  ELECTRICITY,  SOUND,  AND  LIGHT 

one  another's  effects  at  p5,  six  sevenths  at  p7,  etc.  Hence  it  is  not 
necessary  to  go  a  great  distance  above  o  in  order  to  reach  a  region 
in  which  there  are  no  points  at  which  there  is  any  appreciable 
disturbance.  Further,  if  we  consider  wave  lengths  which  are 
shorter  and  shorter  in  comparison  with  'ac,  the  points  of  maximum 
disturbance  p%,  p5,  etc.,  draw  closer  and  closer  together,  and  soon 
some  of  them  begin  to  fall  inside  the  limits  of  the  geometrical 
beam,  i.e.  below  the  point  o.  Hence  those  that  are  left  above  o 
are  weaker  and  weaker  members  of  the  series.  It  follows,  there- 
fore, that  when  the  wave  length  becomes  very  short  in  comparison 
with  ac,  the  disturbance  will  have  become  practically  zero  at  a 
very  short  distance  above  the  point  o.  In  other  words,  a  wave 
motion  should  be  propagated  in  straight  lines  through  an  opening, 
or  past  an  obstacle,  and  should  not  bend  around  appreciably  into 
the  region  of  the  geometrical  shadow,  when  and  only  when  the  wave 
length  is  very  minute  in  comparison  with  the  size  of  the  opening ; 
for  in  this  case  the  disturbances  from  the  various  elements  of  the 
opening  must  interfere  in  such  a  way  as  completely  to  destroy 
one  another  at  practically  all  points  outside  the  limits  of  the 
geometrical  beam.  The  analysis  of  the  conditions  which,  exist 
inside  the  limits  of  the  geometrical  beam  when  p^,  p4,  etc.,  fall 
below  o  will  not  here  be  taken  up,  since  we  are  not  concerned 
at  this  point  with  showing  what  happens  inside  of  or  so  much 
as  with  proving  that  practically  nothing  happens  outside  of  or. 
Suffice  it  to  say  that  experiment  and  theory  both  show  that  under 
the  conditions  assumed  there  is  practically  uniform  disturbance 
within  the  region  or. 

Now  since  ordinary  sound  waves  have  a  wave  length  of  from 
1  to  8  feet,  it  will  be  seen  from  the  above  analysis  that  in  passing 
through  a  window  or  any  ordinary  opening  they  may  be  expected 
to  spread  out  in  all  directions  beyond  the  opening,  as  in  fact  we 
know  that  they  do.  Indeed,  if  the  aperture  is  less  than  one 
wave  length  in  width,  it  should  be  impossible  to  find  any  point  of 
quiescence  whatever  on  the  side  of  the  screen  which  is  away  from 
the  source.  It  is  clear,  then,  that  we  must  produce  extremely 
short  sound  waves,  if  we  are  to  hope  to  observe  with  any  ordinary 
openings  the  diffraction  phenomena  presented  in  the  above  theory. 


DIFFKACTION  OF  SOUND  AND  LIGHT  WAVES     245 

In  light,  however,  since,  as  we  shall  presently  see,  the  average 
wave  length  according  to  the  wave  theory  is  only  about  .00005  cm., 
we  should  expect  that  with  ordinary  openings  the  maxima  p8,  p5) 
etc.,  would  lie  so  near  to  the  edges  o  and  r  of  the  geometrical  beam 
as  not  to  be  easily  discernible,  so  that,  in  order  to  bring  them 
into  evidence  at  all,  we  should  expect  to  be  obliged  to  work  with 
exceedingly  small  openings.  There  is  a  still  further  condition,  how- 
ever, which  must  be  met  in  order  to  bring  out  with  maximum 
clearness,  in  the  case  of  either  light  or  sound,  the  diffraction  bands 
Pi>  Ps>  etc-»  of  the  preceding  theory.  It  is  that  the  wave  experi- 
mented upon  be  converging  instead  of  plane  or  diverging,  as 
assumed  in  the  preceding  discussion.  Before  presenting,  therefore, 
experiments  on  diffraction  it  is  desirable  to  consider  the  methods 
by  which  the  form  of  a  wave  may  be  altered. 

155.  Formation  of  images  by  changes  in  wave  form.  For  the 
sake  of  clearness  let  us  take  a  concrete  case  and  imagine,  with 
Huygens,  that  light  is  a  wave  motion,  and  that  a  light  wave 
originating  in  a  point  S  E 

(Fig.  168)  beneath  a  sur-  v<i>/ 

face  of  water  spreads 
from  that  point  as  a 
spherical  wave.  Let  us 
imagine  further  that 
light  travels  faster  in 
air  than  it  does  in  water. 
Then  it  can  easily  be 
seen  that  the  wave  will 
undergo  a  change  in 

curvature  in    passing 

,,  ,       ,,         .    ,     ,  FIG.  168 

through    the    interlace 

mn  into  air.  For  if  there  were  no  change  in  the  velocity  of  prop- 
agation in  going  from  water  into  air,  then  the  wave  front  which 
at  one  instant  had  reached  the  position  cod  would  an  instant  later 
have  reached  the  position  mo'n,  so  drawn  that  cm  =  oo'  =  dn.  But 
if  oo"  represents  the  distance  which  light  travels  in  air  while  it  is 
traveling  the  distance  oo'  in  water,  then  the  wave,  upon  emergence 
into  air,  should  occupy  some  position  mo"n  instead  of  mo'n.  In 


246 


ELECTKICITY,  SOUND,  AND  LIGHT 


other  words,  the  waves  should  bulge  upward  upon  passing  from 
the  water  to  the  air,  so  that  an  eye  placed  at  E  should  receive  a 
portion  of  a  wave  which  actually  has  its  center  at  some  point  8' 
instead  of  at  S.  This  means,  of  course,  that  the  point  S  should  ap- 
pear to  be,  not  at  its  real  position,  but  at  some  higher  position  S'. 
This  is  precisely  what  everyday  experience  teaches  is  the  case ;  for 
objects  under  water  always  appear  closer  to  the  surface  than  they 
actually  are.  The  point  Sf  is  called  the  image  of  the  point  S.  It  is 
further  called  a  virtual  rather  than  a  real  image,  because,  although 
the  center  of  the  wave  which  reaches  the  eye  is  at  Sf,  the  actual 
center  of  disturbance  is  at  some  other  point,  namely  at  S. 

Again,  when  a  spherical  wave  from  a  point  S  strikes  the  bound- 
ary mn  (Fig.  169)  of  a  new  medium,  it  is  to  be  expected  that 

a  portion  of  it  will  undergo 
reflection,  and  that,  at  the 
instant  at  which  the  ad- 
vancing wave  front  would 
have  been  in  the  position 
mo'n,  had  there  been  no  re- 
flection, the  reflected  wave 
will  actually  be  in  the  posi- 
tion of  the  circular  arc  mo"n, 
so  drawn  that  oo"  =00'.  This 
means,  of  course,  that  an  eye 
at  E,  or  at  any  point  above 
mn,  would  receive  a  wave 
which  has  its  center  at  Sr 
instead  of  at  S.  As  above, 
1<'IG.  109  Sf  is  called  the  image  of  S. 

In  general,   when  for   any 

reason  a  spherical  wave  originating  at  a  point  in  space  is  so 
modified  that  the  wave  appears  to  have  originated  at  a  different 
point,  this  second  point  is  called  the  image  of  the  first  point.  It 
will  be  obvious  that  in  this  case,  as  in  that  preceding,  S'  is  a 
virtual  image  of  S. 

Since  in  Figure  169  the  chord  mn  is  common  to  the  two  arcs 
mo'n  and  mo"n,  it  is  evident  that  the  image  of  a  point  in  a  mirror 


DIFFRACTION  OF  SOUND  AND  LIGHT  WAVES     247 


FIG.  170 


is  on  the  perpendicular  drawn  from  the  point  to  the  mirror,  and 
as  far  behind  the  mirror  as  the  point  is  in  front  of  it.  The  figure 
also  shows  that  the  light  which  comes  to  the  eye  by  reflection  from 
the  mirror  must  follow  the  law,  angle  of  incidence  Srp  =  angle  of  re- 
flection prE;  for  since  oS  =  oSr,  Z  rSo  =  Z  rS'o  —  /_prS  =  /.prE. 

These  theoretical  deductions  D 
from  the  wave  theory  are  pre- 
cisely the  laws  which  experi- 
ment shows  to  be  those  which 
govern  the  reflection  of  light 
from  plane  mirrors. 

A  third  method  of  modify- 
ing the  shape  of  a  wave,  and 

one  which  finds  large  application  in  optical  instruments,  consists 
in  causing  the  wave  to  pass  through  a  lens  in  the  manner  shown 
in  Figure  170.  If  the  speed  of  propagation  of  the  wave  is  less  in 
the  material  of  the  lens,  for  example  glass,  than  it  is  in  air,  then, 
since  the  portion  of  the  wave  which  passes  through  the  middle  of 
the  lens  is  retarded  more  than  the  portions  which  pass  through  the 
edges,  it  is  obvious  that  the  lens  will  tend  to  reverse  the  direction 
of  curvature  of  the  wave.  If  the  source  S  is  close  to  the  lens,  the 
curvature  of  the  wave  front  which  has  passed  through  the  lens 
may  not  be  reversed,  but  it  will  be  diminished ;  that  is,  the  wave 
will  actually  be  flattened  so  that  its  center  will  lie  to  the  left  of  S. 
In  this  case  an  eye  placed  to  the  right  of  the  lens  and  looking 

toward  S  will  see  a  virtual 
image  of  S  at  the  point  p  at 
which  this  center  is  located 
(see  Fig.  170).  If  the  distance 
of  S  from  the  lens  is  that  of 
the  so-called  principal  focal 
plane  F,  the  emerging  wave 
will  be  plane  (Fig.  171);  but  if  S  is  to  the  left  of  the  principal 
focal  plane  F,  the  emerging  wave  will  be  reversed ;  that  is,  it  will 
be  concave  toward  the  direction  in  which  it  is  traveling  (Fig.  172). 
It  cannot,  in  general,  be  assumed  that  the  emerging  wave  front 
will  be  strictly  spherical,  but  if  a  diaphragm  ac  is  introduced,  as 


FIG.  171 


248 


ELECTRICITY,  SOUND,  AND  LIGHT 


in  Figure  172,  so  as  to  cut  out  those  portions  of  the  wave  which 
have  passed  through  the  edges  of  the  lens,  the  remaining  wave 
front  may,  with  a  suitable  lens,  be  made  practically  spherical.  In 
this  case  the  disturbances  starting  from  every  point  on  the  spher- 
ical surface  reach  the  center  p  of  the  sphere  at  precisely  the  same 
instant.  Since  these  disturbances  will  all  be  in  the  same  phase, 


s-  •  I) 


(  i  > 


FIG.  172 

they  will  unite  to  produce  a  disturbance  of  very  great  intensity 
at  p.  In  ordinary  language  the  light  from  S  will  be  focused  at  p. 
In  technical  terms  a  real  image  of  S  will  be  formed  at  p.  Such 
an  image  is  distinguished  from  the  virtual  images  which  have 
thus  far  been  discussed  in  that  the  point  p  now  becomes  an  actual 

P 


FIG.  173 

center  of  disturbance  from  which  waves  spread  out  to  the  right 
of  p  as  though  the  source  S  were  itself  at  this  point  (see  Fig.  172). 
A  second  method  of  producing  a  wave  which  is  concave  toward 
the  direction  in  which  it  is  traveling  is  to  allow  a  diverging  wave 
of  suitable  curvature  to  be  reflected  from  a  concave  mirror.  If  the 
source  S  (Fig.  173)  is  closer  to  the  mirror  than  the  principal  focal 


DIFFRACTION  OF  SOUND  AND  LIGHT  WAVES     249 


plane  F,  the  reflected  wave  will  still  be  convex  toward  the  direc- 
tion in  which  it  is  traveling,  as  is  shown  in  the  figure  which  is 
constructed  precisely  as  was  Figure  169,  oo"  being  made  equal  to  oo'. 
But  if  S  is  farther  from  the  mirror  than  the  principal  focal  plane, 
then  the  reflected  wave  will  be  concave,  and  a  real  image  will  be 
formed  at  p  (see  Fig.  174).  In  the  cases  illustrated  in  Figures  170, 
172, 173,  and  174,  the 
points  S  and  p  are  ' '  •  p  •  -  i  i  j  | 
called  conjugate  foci. 
In  Figure  171  the  con- 
jugate focus  is  obvi- 
ously at  infinity. 

156.  The  nature  of  a  real  image  of  a  point  source.  An  appli- 
cation of  the  reasoning  of  section  154  to  the  case  of  a  converging 
wave  shows  that  the  image  of  a  point  source  of  waves  should 
not  be  a  single  point  of  disturbance,  but,  instead,  a  series  of  max- 
ima and  minima  of  disturbance.  For  consider,  as  in  section  154, 
a  point  p2  (Fig.  175)  far  enough  to  one  side  of  p  so  that  apz  is 
one  wave  length  more  than  cpz.  The  disturbance  from  the  portion 


*/>„ 


FIG.  174 


FIG. 175 

ab  of  the  wave  front  dbc  will  completely  neutralize  at  pz  the  dis- 
turbance from  the  portion  be.  Furthermore,  the  contrast  between 
the  disturbance  at  p  and  the  absence  of  disturbance  at  p2  will 
be  much  more  pronounced  in  this  case  than  it  was  in  the  case 
illustrated  in  Figure  167,  since  in  the  case  of  Figure  175  all  of 
the  disturbances  from  the  wave  front  ac  are  concentrated  at  the 
center  p,  thus  making  it  a  point  of  extreme  brightness,  while  in 
the  case  illustrated  in  Figure  167  no  such  concentration  occurs. 


250  ELECTRICITY,  SOUND,  AND  'LIGHT 

Precisely  as  in  Figure  167,  at  p3,  p&,  and  other  points  for  which 
the  distance  from  a  to  the  point  in  question  is  any  odd  multiple 
of  a  half  wave  length  greater  than  the  distance  from  c  to  this 
point,  there  will  be  maxima  of  disturbance ;  but  these  maxima 
will  decrease  rapidly  in  intensity,  ps  being  about  a  sixtieth  as 

.  intense  as  p,  p5  one  two  hundredth  as  intense  as  p,  etc.  Moreover, 
p3,  p5,  etc.,  will  draw  very  close  together  and  very  close  to  p  as 
the  wave  length  \(=ad)  becomes  small  in  comparison  with  the 

•  aperture  ac.  In  other  words,  the  distance  apart  of  the  maxima 
p,  ps)  etc.,  will  depend  simply  upon  the  ratio  between  the  wave 
length  X  and  the  aperture  ac. 

If,  then,  as  the  wave  theory  demands,  the  wave  length  of  light 
is  very  minute  as  compared  with  the  diameters  of  ordinary  lenses 
or  mirrors,  the  maxima  pa,  p5,  etc.,  will  be  so  close  to  p  as  to  be 
indistinguishable  from  it;  so  that  for  practical  purposes  common 
lenses  or  mirrors  should  form,  in  the  case  of  light,  essentially 
point  images  of  point  sources,  and  at  all  points  outside  the  limits 
of  the  geometrical  beam,  that  is  at  all  points  not  included  within 
the  region  inclosed  by  the  lines  ap  and  cp,  the  light  waves  should 
mutually  destroy  one  another. 

Nevertheless,  that  the  image  of  a  point  is  not  in  reality  a  point 
even  in  the  case  of  light  waves,  but  consists  of  a  series  of  maxima 
and  minima,  as  the  preceding  theory  demands,  may  be  easily 
demonstrated  by  reducing  the  opening  ac  until  it  becomes  more 
nearly  comparable  with  a  light  wave.  Thus  a  pin  hole  in  a  piece 
of  cardboard  held  immediately  in  front  of  a  bright  flame  and 
viewed  at  a  distance  of  two  or  three  feet  through  a  small  pin  hole 
in  another  card  held  very  close  to  the  eye  will  appear,  not  as  a 
point,  but  as  a  central  bright  disk  surrounded  by  one,  two,  or 
even  more  black  rings  which  correspond  to  the  points  pz,  p±,  etc., 
of  Figure  175.  Again,  if  a  slit,  say  a  half  millimeter  wide,  be 
made  to  replace  the  first  pin  hole,  and  if  it  be  viewed  at  a  dis- 
tance of  a  few  feet  through  another  slit,  say  one  tenth  millimeter 
wide,  which  is  held  very  close  to  the  eye,  the  first  slit  will  appear 
as  a  central  bright  band  flanked  by  a  series  of  dark  bands  which 
correspond  to  the  points  p9,  p4,  etc.,  of  Figure  175.  In  these  experi- 
ments the  remote  pin  hole  or  slit  corresponds  to  the  point  S,  the 


DIFFRACTION  OF  SOUND  AND  LIGHT  WAVES     251 


lens  of  the  eye  to  L,  the  retina  of  the  eye  to  the  screen  upon 
which  the  points  p,  pv  p.z,  ps,  etc.,  are  observed,  and  the  pin  hole  or 
slit  held  very  close  to  the  eye  to  the  aperture  ac,  which  is  indeed 
in  this  case  on  the  other  side  of  the  lens,  but  this  fact  does  not 
alter  in  any  way  the  theory  of  the  phenomenon.  The  experiment 
may  be  made  more  striking  by  throwing  a  beam  of  direct  sunlight 
through  a  half-millimeter  slit  covered  with  red  glass,  and  then 
observing  the  slit  through  a  telescope  placed  in  the  path  of  the 
beam.  So  long  as  the  lens  has  its  normal  aperture  the  image  of 
the  slit  as  seen  in  the  telescope  will  be  sharply  denned,  if  the  tele- 
scope is  properly  focused.*  But  when  the  aperture  is  made  small 
by  slipping  a  second  slit  over  the  objective  of  the  telescope  parallel 
to  the  first,  the  series  of  light  and  dark  bands  p,  p<,,  p3,  etc.,  will 
at  once  appear.  Furthermore,  as  the  width  of  the  slit  over  the 
objective  is  made  larger  and  larger  these  bands  will  draw  closer 

*  The  way  in  which  a  lens  or  mirror  forms  an  image  of  an  extended  object, 
such  as  a  slit,  may  be  readily  seen  by  regarding  the  object  as  an  assemblage 
of  points.  If  the  diameter  of  the  lens  or  mirror  is  very  large  in  comparison 
with  a  wave  length,  each  point  A  or  B  (Fig.  176)  of  the  object  will  have  what 
is  practically  a  point  image  of  itself  formed  at  a  or  6.  Further,  the  points  a 
and  b  will  lie,  approximately  at  least,  in  the  prolongations  of  the  lines  drawn 
from  A  and  B  respectively  to  the  center  C  of  the  lens,  as  is  evident  from  the 
approximate  symmetry  of  the  lens,  and  hence  of  the  retardation  produced 
in  the  incident  wave  by 
it,  about  the  lines  AC  /] 
and  BC.  Similarly,  the 
image  of  any  point  on 
the  object  will  be  in  the 
prolongation  of  the  line 
connecting  this  point  with 
the  center  of  the  lens  or 
mirror  (see  also  Figs. 
170  and  173). 

When,  however,  we  consider  the  clearness  with  which  detail  in  the  object 
is  brought  out  in  the  image,  we  must  even  here  remember  that  the  images  of 
points  are  not  points  but,  instead,  bright  centers,  each  of  which  is  surrounded 
by  its  series  of  bright  and  dark  rings.  If  two  of  these  centers  are  closer  together 
than  the  radius  of  the  first  dark  ring  about  each,  they  will  evidently  not  be  sep- 
arated in  the  image  by  a  dark  region,  and  will  therefore  not  be  distinguishable 
as  separate  points.  That  is,  a  lens  cannot  produce  an  image  which  will  make 
two  close  bright  points  appear  as  separate  points  if  the  angle  subtended  at  the 
lens  by  the  two  points  is  less  than  X/ac.  This  expression  X/ac,  or  the  wave 
length  divided  by  the  aperture,  is  therefore  known  as  the  limit  of  resolution  of 
the  lens. 


FIG.  170 


252  ELECTRICITY,  SOUND,  AND  LIGHT 

and  closer  together  and  finally  be  lost  in  the  central  image  p  of 
the  first  slit,  precisely  as  the  above  theory  demands  that  they 
should. 

In  order  to  demonstrate  the  existence  of  the  same  phenomena 
with  sound  waves  it  is  obvious  that,  if  the  mirrors  or  lenses  are 
to  be  of  ordinary  size,  we  must  find  a  way  of  producing  sound 
waves  of  unusual  shortness.  Such  waves  may  be  produced  by  the 
whistle  described  in  the  next  section.  They  must  be  detected  and 
measured,  however,  by  a  special  appliance  called  a  sensitive  flame. 

157.  The  sensitive-flame  apparatus.  The  sensitive  flame  is  a 
very  high  but  very  narrow  flame  produced  by  igniting  a  jet  of  gas 
as  it  issues  from  a  tank  under  high  pressure.  The  burner  has  a 
pin-hole  opening  and  may  be  conveniently  constructed  by  draw- 
ing out  a  piece  of  glass  tubing  with  a  long  taper  until  the  orifice 
has  a  mean  diameter  of  about  half  a  millimeter.  The  opening  is 
in  general  made  slightly  elliptical  and  the  flame  is  most  sensitive 
when  the  shorter  axis  of  the  ellipse  lies  in  the  direction  of  the 
approaching  disturbance.  The  entering  gas  is  regulated  until  there 
results  a  long  flame  usually  twenty  or  thirty  centimeters  high, 
which  is  just  on  the  point  of  flaring. 

Small  disturbances  of  high  frequency  produced  in  the  air  sur- 
rounding the  mouth  of  the  burner  cause  the  flame  to  flare  and 
shorten  very  perceptibly,  the  reason  being  that  very  sudden  vari- 
ations in  the  pressure  at  the  orifice  of 
Compressed  ^ne  burner,  such  as  are  produced  by 
Air  very  rapid  vibrations,  cause  correspond- 
ing variations  in  the  rate  of  emission  of 
gas,  and  hence  corresponding  variations 

in  the  height  of  the  flame.  The  flame  is  very  sensitive  to  exceed- 
ingly short  waves  such  as  are  produced,  for  example,  by  jingling 
a  bunch  of  keys.  To  the  ordinary  notes  of  audible  sound  it  is 
wholly  insensitive,  because  such  notes  do  not  produce  sufficiently 
sudden  changes  in  pressure  at  the  orifice  of  the  burner.  There 
is  used  with  the  flame,  therefore,  as  a  source  of  waves,  a  small 
whistle  shaped  as  shown  in  cross  section  in  Figure  177,  the  dis- 
tance between  the  two  small  openings  being  from  one  to  three 
millimeters.  The  whistle  is  blown  by  compressed  air  and  acts  as 


DIFFRACTION  OF  SOUND  AND  LIGHT  WAVES     253 

a  very  short  organ  pipe,*  the  emission  of  pulses  from  each  of  the 
two  openings  being  controlled  by  the  period  of  the  reflected  pulses 
which  bound  back  and  forth  between  the  two  ends  of  the  minute 
air  chamber.  Ordinarily  no  recognizable  note  is  heard,  for  the 
reason  that  the  note  produced  is  above  the  limits  of  audition. 


EXPERIMENT  21 

(A)  Object.  To  show  experimentally  that  with  sound  waves  of  very  short 
wave  length  sharp  shadows  are  cast  by  ordinary  objects. 

Directions.  Set  up  the  sensitive  flame  described  in  section  157.  Turn 
on  the  gas  until  the  flame  is  from  ten  to  fourteen  inches  high  and  just  on 
the  point  of  flaring,  and  also  until  a  slight  rattling  of  a  bunch  of  keys  will 
cause  it  to  flare  strongly.  Locate  the  side  of  the  flame  which  is  most  sen- 
sitive to  this  noise,  and  in  all  future  use  of  the  flame  turn  that  side  toward 
the  direction  from  which  the  disturbance  comes.  , 


FIG.  178 

Set  up  the  whistle  W  in  the  manner  shown  in  Figure  ITS,  so  that  it  is 
in,  or  just  a  trifle  beyond,  the  principal  focal  plane  F  of  a  concave  mirror 
of  about  twenty  centimeters  focal  length  f  and  twenty  centimeters  aperture 

*  A  constant  air  pressure  is  desirable,  since  otherwise  overtones  varying  with 
the  changes  of  pressure  will  be  produced.  A  pressure  of  about  15  cm.  of  water 
is  usually  sufficient. 

t  If  the  focal  length  .F  of  the  mirror  is  not  known,  and  if  the  latter  is  made 
of  plaster  of  Paris,  or  of  some  other  material  which  renders  a  determination  by 
optical  means  impossible,  then  determine  F  by  measuring  d  and  ac  (Fig.  178), 
and  substituting  in  the  formula  F  —  ac2/l6  d.  This  formula  is  obtained  from  a 
consideration  first  of  the  fact  that  ac/2  is  a  mean  proportional  between  d  and 
the  other  segment  of  the  diameter  of  the  circle  of  which  the  mirror  is  an  arc ; 
second,  of  the  fact  that,  since  d  is  very  small,  this  other  segment  may  be  taken 
as  the  diameter  itself  ;  and  third,  of  the  fact  that  the  focal  length  of  the  mirror 
is  one  half  its  radius  of  curvature  (see  Chap.  XXIII). 


254 


ELECTRICITY,  SOUND,  AND  LIGHT 


(ac,  Fig.  178).  Place  the  sensitive  flame  G  in  the  principal  axis*  of  the 
mirror  and  at  a  distance  from  it  of  about  eight  feet.  Slowly  increase  the 
air  pressure  with  which  the  whistle  is  to  be  operated  until  the  flame  begins 
to  flare  strongly.  Turn  the  tripod  rod  R  which  supports  the  mirror  and 
whistle  about  a  vertical  axis  and  observe  that  under  the  conditions  pre- 
scribed a  sharply  outlined  beam  of  sound  is  produced  which  causes  the 
flame  to  flare  when  it  strikes  it,  but  leaves  it  burning  undisturbed  when 
turned  in  another  direction. 

Turn  the  beam  again  upon  the  flame  ;  then  interpose  between  it  and  the 
whistle  a  strip  of  cardboard,  or  some  other  object  a  few  centimeters  wide, 

and  observe  that  when  in  the 
right  position  it  causes  the 
flaring  to  cease,  thus  showing 
that  with  these  short  sound 
waves  even  small  objects  cast 
sharp  shadows. 

Next  let  a  beam  of  sound 
SC  (Fig.  179)  be  sent  from 
the  mirror  through  an  opening 
about  a  foot  in  diameter  be- 
tween two  screens  db  and  ce?, 
disposed  as  in  the  figure,  and 
let  this  beam  be  reflected  by 
a  flat  smooth  board  C  to  the 
sensitive  flame  placed  at  F.  It 
will  be  found  that  the  flame 
will  flare  strongly  only  when 
the  board  is  so  turned  that  the 
FIG.  179  angle  of  incidence  SCNis  equal 

to  the  angle  of  reflection  NCF. 

(B)  Object.  To  find  the  wave  length  of  the  note  emitted  by  the  whistle 
by  locating  nodes  and  loops  in  front  of  a  reflecting  wall. 

Directions.  Set  up  a  smooth  planed  board  behind  the  flame  G,  arranged 
as  in  Figure  178,  and  let  the  plane  of  the  board  be  at  right  angles  to  the 
line  connecting  the  flame  and  whistle.  Attach  a  base  to  the  board  so  that, 
as  the  latter  is  moved  back,  its  plane  remains  parallel  to  itself.  Starting 
with  the  board  but  a  centimeter  or  so  from  the  flame,  move  it  a  few  milli- 
meters forward  or  back  until  a  position  is  found  in  which  the  flame  ceases 
to  flare.  Mark  the  position  of  the  board  with  a  piece  of  chalk ;  then  slide 
it  slowly  back  from  the  flame  and  count  the  nodes,  i.e.  the  positions  of  no 
flaring,  as  each  is  passed.  Mark  the  position  of,  say,  the  twenty-fifth  node 


*  The  principal  axis  is  a  line  drawn  through  the  center  of  the  mirror  and  its 
center  of  curvature  C. 


DIFFRACTION  OF  SOUND  AND  LIGHT  WAVES     255 


and  find  the  wave  length  of  the  note  given  off  by  the  whistle  from  a  consid- 
eration of  the  fact  that  the  distance  between  successive  nodes  is  one  half 
wave  length. 

(C)  Object.  To  find,  by  means  of  diffraction  experiments,  the  wave 
length  of  the  note  used  in  (B). 

Directions.  Place  the  whistle  W  (Fig.  178)  in  a  position  on  the  princi- 
pal axis  of  the  mirror  2  or  3  cm.  from  its  principal  focus.  Then  from  the 
distance  /(=  MS,  Fig.  180)  from  this  point  to  the  mirror,  and  the  focal 
length  F,  compute  the  distance  /'  (  =  pM,  Fig.  180)  from  the  mirror  to  the 
conjugate  focus  p  by  means  of  the  formula 

1+1=1* 
/  /-    f 

Place  the  mirror  and  whistle  so  that  the  orifice  of  the  flame  G  (Fig.  178) 
is  accurately  at  this  conjugate  focus  p]  then  slowly  rotate,  about  its  own 
axis,  the  vertical  rod  R  which  carries  the  mirror  and  whistle,  and  thus 
locate,  on  either  side  of  the  central  region  p  (Fig.  180)  of  intense  disturb- 
ance, at  least  one  point  p.2  of  no  disturbance,  and  one  more  point  p8  of 


FIG.  180 


maximum  disturbance.  Attach  a  horizontal  index  of  length  I  (for  exam- 
ple, 50  cm.)  to  the  vertical  rod  R  and  measure  the  length  of  arc  a  traced 
out  by  the  end  of  this  index  as  the  mirror  is  rotated  from  the  position  in 
which  the  flame  is  burning  quietly  at  />„  (Fig.  180)  to  the  position  in 
which  it  is  burning  quietly  at  the  point  p<,  which  is  on  the  other  side  of  p. 
It  will  be  evident  at  once  from  Figure  180  that  the  angle  through  which 
the  mirror  must  be  rotated  to  cause  the  point  p  to  move  over  to  the  posi- 
tion of  p2  is  pp2/pM.  But  this  angle  is  also  \/ac  (see  Fig.  180).  Hence 
the  equation 


pM 


But  obviously, 


Hence 


2  X 

- 
ac 


PiPi  _  ^ 


or 


pM 


la 

\=--ac. 
11 


*  See  Chapter  XXIII  for  the  derivation  of  this  formula. 


256  ELECTRICITY,  SOUND,  AND  LIGHT 

Compare  the  value  of  X  thus  found  with  the  result  obtained  in  (B).* 
(D)  Object.    To  determine,  by  diffraction  experiments,  the  average  wave 
length  of  red  light. 

Directions.  At  a  distance  of  about  3  in.  from  a  vertical  slit  s  (Fig.  181) 
about  .5  mm.  wide,  set  up  an  ordinary  reading  telescope  T.  Throw  a 
beam  of  sunlight  through  s  in  the  direction  sT,  cover  s  with  a  piece 
of  red  glass  so  that  only  red  light  reaches  the  telescope  T  from  s  ;  then 
focus  T  carefully  upon  the  slit  s,  and  thus  obtain  a  sharply  defined  image 
of  s  in  the  eyepiece  of  T.  Now  place  over  the  objective  of  T  a  cap  con- 
taining a  slit  not  more  than  .5  mm.  wide,  and  rotate  this  slit  until  it  is 
parallel  to  the  slit  s.  We  have  then  precisely  the  conditions  given  in 
Figure  175  for  the  formation  of  diffraction  bands.  In  a  manner  identical 
with  that  used  in  (C)  we  may  now  turn  the  telescope  T  on  its  vertical  sup- 
port and  observe  the  angle  p2p$/pL  (Fig.  175)  through  which  it  must  pass  in 
order  to  cause  the  cross  hairs  to  move  from  one  of  the  two  dark  bands  p2, 
adjacent  to  the  central  bright  one  p,  to  the  other.  In  order  to  measure  this 


angle  accurately,  attach  to  the  telescope  support  a  mirror  m  and  observe 
its  rotation  with  a  second  telescope  and  scale  t  (see  Fig.  181).  If  d  is  the 
deflection  observed  on  the  scale  of  t,  and  if  D  is  the  distance  from  the  mirror 
to  the  scale,  then  d/2  D  is  obviously  the  angle  sought.  To  obtain  as  accu- 
rate a  determination  as  possible  of  this  angle,  take  the  mean  of  the  distances 
P^Pzi  P±Pi/~  anc^  PsPQ/S-  Measure  with  a  micrometer  microscope  the  width 
ac  of  the  slit  over  the  objective,  then  compute  the  wave  length  A,  of  red  light 

from  the  relation 

A         d 


and  compare  the  result  thus  found  with  the  value  .000067  cm.,  which  is 
about  the  mean  value  of  the  wave  lengths  of  the  light  transmitted  by 
ordinary  red  glass. 

*  Rigorously,  the  analysis  given  in  section  150  applies  only  to  a  single  plane, 
that  is,  to  a  rectangular  aperture  or  slit,  the  dimension  of  which  at  right  angles 
to  the  plane  of  the  paper  is  very  great  in  comparison  with  the  dimension  ac. 
The  complete  analysis  for  a  circular  aperture  such  as  that  here  used  is  too 
complicated  to  be  within  the  scope  of  this  book.  Suffice  it  to  say.  that  such 
analysis  leads  to  the  result  that  pp2  for  circular  apertures  is  1.22  times  its 


DIFFRACTION  OF  SOUND  AND  LIGHT  WAVES     257 

EXAMPLE 

-     (A)  The  predictions  as  to  sharp  shadows  formulated  in  the  directions 
were  verified  by  experiment. 

(B)  By  the  method  of  locating  nodes  in  front  of  a  reflecting  wall  twenty 
half  wave  lengths  were  found  to  be  equal  to  17  cm.    Hence  A.  =  .85  cm. 

(C)  The  wave  length  was  determined  by  the  method  of  diffraction  as 
follows.    The  diameter  ac  of  the  mirror  was  20.4  cm.,  the  sagitta  d  was 
1.30  cm.    Hence  R  =  To722/2  (1.3)  =  40  cm.  and  F  =  20  cm.     The  whistle 
was  fastened  to  a  rod  attached  to  the  vertical  axis  with  which  the  mirror 
rotated.   It  was  placed  a  distance  /  of  22.1  cm.  from  the  center  of  the  mirror 
along  the  principal  axis.    The  sensitive  flame  was  placed  at  a  distance/'  of 
210  cm.  measured  along  this  axis  from  the  mirror,  the  value  of  f  having 
been  found  by  the  relation  1/22.1  +  I/  'f  =  1/20.    An  index  47  cm.  long  was 
fastened  to  the  vertical  rod  and  turned  with  the  mirror.    As  the  mirror 
turned  through  the  angle  between  p2  and  p2  the  end  of  the  index  moved 

over  4.65  cm.    Hence  (see  note,  page  256)    X  =  -  •  —  —  •  —  —  =  .83  cm. 

This  value  agreed  with  that  of  (B)  to  within  2.4  per  cent. 

(D)  Four  dark  bands  on  either  side  of  the  central  bright  image  were 
easily  observed  through  an  aperture  of  width  ac  =  .0404  cm.    A  series  of 
readings  was  taken  on  all  these  bands  with  a  telescope  and  scale  placed 
162  cm.  from  the  mirror  ra  attached  to  the  first  telescope  T.     The  differ- 
ences in  readings  on  this  scale  for  settings  on  the  bands  were  as  follows. 
From  p2  to  p2,  1.09cm.;  from  jo4  to  p4,  2.16  cm.;  from  pe  to  j»0,  3.30cm.; 
and  from  p8  to  /?8,  4.40  cm.     Hence  the  average  distance  on  the  scale  cor- 
responding to  the  angle  between  p2  and  p2  was  1.095  cm.    Hence,  since 


X  =  ,  we  have  X  =      -  =  .0000685.    This  is  in  agreement 

with  the  mean  value  of  .000067  to  within  1.7  per  cent. 

value  for  a  rectangular  opening.  Hence  the  value  of  X  found  as  above  must 
be  divided  by  1.22  in  order  to  obtain  an  accurate  comparison  with  the  value 
found  in  (B).  Rigorously,  then,  we  have 


~2  pM 


CHAPTER  XXII 


THE   DIFFRACTION   GRATING 

158.  The  principle  of  the  grating.  The  phenomena  of  diffrac- 
tion are  most  strikingly  exhibited  with  the  aid  of  an  instrument 
devised  in  "1821  by  the  celebrated  German  optician  Fraunhofer, 
and  known  as  the  diffraction  grating.  Such  a  grating  consists 
essentially  of  an  opaque  screen  in  which  are  placed  at  regular  in- 
tervals small  parallel  slits  for  the  transmission  or  reflection  of 
light.  Thus  in  Figure  182  qq'  represents  a  cross  section  of  the 
grating,  and  the  openings  m,  p,  s,  etc.,  represent  the  slits,  which 

are  thought  of  as 
extending  at  right 
angles  to  the  plane 
of  the  page.  For 
the  sake  of  con- 
venience in  analy- 
sis these  slits  will 


at  first  be  regarded 
as  exceedingly  nar- 
row in  comparison 
with  their  distance 
apart,  that  is,  each 
opening  will  be 
thought  of  as  a  mere  line.  Let  the  source  of  light  be  so  far  distant 
that  the  wave  surface  ww1  which  falls  upon  the  grating  is  practically 
plane.  It  has  been  shown  in  the  preceding  chapter  that  if  no  grat- 
ing were  present,  a  lens  L  interposed  in  the  path  of  the  wave  ww' 
would  form  an  image  of  the  distant  source  S  at  some  point  a,  while 
at  all  other  points  on  the  screen  uv  there  would  be  destructive 
interference  and  therefore  total  darkness.  Let  us  see  how  this  con- 
clusion would  be  modified  if  we  take  out  certain  portions  of  the  wave 

258 


FIG.  182 


THE  DIFFRACTION   GRATING  259 

front  ww'  by  means  of  the  grating.  When  the  wave  ww'  reaches 
the  grating  qq'  the  points  m,  p,  s  become  new  sources  of  spherical 
waves,  and  if  we  draw  the  envelope  to  all  these  waves  after  the 
disturbance  has  traveled  a  small  distance  forward,  we  shall  obtain 
precisely  the  same  surface  AB  which  we  should  have  had  if  the 
grating  had  not  been  present,  the  only  difference  being  that  the  in- 
tensity of  disturbance  in  the  plane  AB  is  much  less  than  before, 
since  now  but  a  few  points,  namely  m,  p,  s,  etc.,  are  sending  out 
spherical  waves  to  AB,  while  before  all  the  points  in  qq'  were  so 
doing.  As  was  shown  in  section  155,  the  lens  will  take  this  plane 
wave  AB,  consisting  of  vibrations  all  of  which  are  in  the  same 
phase,  and  bring  it  to  a  focus  at  a,  so  that  an  enfeebled  image  of 
the  distant  source  S  will  be  formed  at  this  point.  Thus  far,  then, 
the  only  effect  of  the  grating  has  been  to  diminish  the  intensity 
of  the  image  at  a. 

But  AB  is  not  now  the  only  surface  wiiich  can  be  drawn  to  the 
right  of  the  grating  so  as  to  touch  points  all  of  which  are  in  the 
same  phase  of  vibration,  for  a  surface  mD,  so  taken  that  the  dis- 
tance from  p  to  it  is  one  wave  length,  that  from  s  two  wave  lengths, 
and  so  on,  satisfies  this  condition  quite  as  well  as  does  the  surface 
AB.  Hence  mD  may  be  regarded  as  another  plane  wave,  which, 
after  passage  through  the  lens,  will  be  brought  to  a  focus  at  some 
point  c  in  the  line  drawn  perpendicular  to  mD  through  the  center 
C  of  the  lens,*  in  precisely  the  same  way  in  which  the  plane  wave 
AB  was  brought  to  a  focus  at  a.  It  follows,  then,  that  an  image 
of  the  source  should  be  formed  at  c  as  well  as  at  a.  Precisely  the 
same  line  of  reasoning  will  show  that  another  image  of  the  distant 
source  should  be  formed  at  c'  as  far  below  a  as  c  is  above  it.  But 
a,  c,  and  cf  are  not  the  only  points  at  which  images  of  the  source 
will  be  formed,  for  it  is  possible  to  pass  a  plane  through  m  such 
that  the  distance  from  p  to  this  plane  is  2  X  instead  of  X,  that 
from  s,  3  X,  etc.  It  is  obvious  that  all  points  in  this  plane  will  be  in 
the  same  phase  of  vibration,  and  hence  that  the  resulting  plane  wave 
will  be  brought  to  a  focus  at  some  point  d  on  the  perpendicular 
drawn  from  the  plane  through  C,  and  at  the  same  distance  from  C 
as  are  a  and  c. 

*  See  footnote  on  page  251. 


260  ELECTKICITY,  SOUND,  AND  LIGHT 

Similarly,  there  will  be  other  images  whose  direction  from  C  is 
determined  by  the  simple  condition  that  the  successive  distances 
from  the  slits  to  the  wave  front  differ  by  a  whole  multiple  of  a 
wave  length  ;  thus  po  =  \,  2  X,  3  X,  4  X,  5  X,  etc.  The  first  image, 
namely  that  at  c,  where  this  difference  is  one  wave  length,  is 
called  the  image  of  the  first  order.  Similarly,  that  at  d  is  the  image 
of  the  second  order,  and  so  on.  In  a  word,  then,  a  lens  and  grating 
disposed  as  in  Figure  182  should  produce  a  whole  series  of  equidis- 
tant images  of  any  distant  source  of  light.  This  means,  of  course, 
that  under  these  conditions  light  waves  will  bend  far  around  into 
the  region  of  the  geometrical  shadow  and  be  discernible  at  a  large 
number  of  different  points  instead  of  simply  at  a. 

These  theoretical  deductions  from  the  wave  theory  of  light  are 
completely  confirmed  by  experiment.  Furthermore,  the  experi- 
ments illustrating  them  are  so  simple  and  so  much  a  part  of 
everyday  experience  that  the  wonder  is  that  they  escaped  detec- 
tion and  explanation  for  so  long  a  time.  Thus  if  one  looks 
through  a  handkerchief  held  close  to  the  eye  at  a  distant  arc 
light,  gas  flame,  or  bright  star,  one  can  always  see  nine  and  some- 
times as  many  as  eighteen  or  more  images  of  the  light.  These  are 
due  to  the  two  sets  of  gratings  formed  by  the  two  sets  of  threads 
which  run  at  right  angles  to  each  other.  It  is  usually  possible  to 
see  as  many  as  three  distinct  images  by  simply  squinting  at  a  dis- 
tant light  through  the  eyelashes,  which  act  in  this  case  as  a  very 
imperfect  grating.  In  these  experiments  the  retina  of  the  eye  takes 
the  place  of  the  screen  uv,  and  the  lens  of  the  eye  the  place  of  L. 

In  its  simplest  practical  form  the  grating  consists  of  a  plane 
piece  of  glass  upon  which  are  ruled  with  a  diamond  point,  say,  a 
thousand  lines  to  the  centimeter.  The  grooves  cut  by  the  diamond 
point  constitute  the  opaque  spaces  in  the  grating,  for  the  light 
which  falls  upon  these  grooves  is  scattered  in  all  directions,  so 
that  a  negligible  part  of  it  passes  through  in  the  direction  in  which 
the  light  is  traveling.  The  clear  glass  between  the  rulings  corre- 
sponds to  the  openings  m,  p,  s  in  the  screen  of  Figure  182.  If  such 
a  grating  is  held  immediately  before  the  eye  and  a  source  of  mono- 
chromatic light  viewed  through  it,  the  series  of  images  formed  at 
a,  c,  d,  etc.,  on  the  retina  are  apprehended  by  the  observer  as  a 


THE  DIFFKACTION   GEATIXG 


261 


series  of  images  of  the  source  lying  in  the  prolongations  of  the 
lines  aC,  cC,dC,  etc.  The  images  may  be  thrown  upon  a  screen,  if 
screen,  lens,  grating,  and  source  are  given  the  relative  positions 
shown  in  Figure  182. 

159.  The  determination  of  wave  length  by  means  of  the  grating. 

If  a  grating  is  held  immediately  in  front  of  the  eye  and  a  source 
of  monochromatic  light  viewed  through  it,  the  wave  length  of 
the    light   vibration    may  be    determined    as    follows.    In 
Figure  183    let    the   angle   between   the   grating   and  the 


FIG.  183 

direction  of  the  wave  front  mD,  which  forms  the  image  of 
the  first  order,  be  denoted  by  6.     Let  the  wave  length  be 
denoted  by  X,  and  let  d  represent  the  distance  between  successive 
openings, — the  grating  space,  as  it  is  called.    Now  in  the  triangle 
pmh  the  obvious  relation  exists, 

a) 


•      4 

sin  0  —  —  • 
d 


Similarly,  for  the  image  of  the  second  order, 

2JV 
d' 


sin  0'= 


(2) 


And,  in  general,  if  n  represents  the  order  of  any  image  and  6  the 
angle  between  the  grating  and  the  direction  of  the  wave  forming 
that  image,  then  -. 

o        71  A,  ,  n  \ 

sin  0  =  —  •  (3) 


262  ELECTRICITY,  SOUND,  AND  LIGHT 

The  determination  of  the  angle  6  may  be  easily  made  as  follows. 
An  illuminated  slit  is  placed  at  s  (see  Fig.  183)  in  front  of  a  scale 
fg  two  or  three  meters  away  from  the  grating  and  the  eye.  The 
positions  s\  s",  s^,  s"l}  etc.,  at  which  the  successive  images  formed 
by  the  grating  appear,  are  then  marked  in  some  way  upon  the 
scale  fg.  If  the  grating  has  been  placed  so  that  the  line  of  sight 
sE  is  perpendicular  to  it,  then,  since  s'E  is  perpendicular  to  the 
wave  front  mD,  we  have  the  angle  sEs1  =  6,  and  sin  0  =  ss'/s'E. 

Therefore  ~  ,  , 

X       ss'  ,  ss' 

-  =  _,       or       X  =  rf  — •  (4) 

The  distance  s'E  can  easily  be  measured  with  a  tape.  To  obtain 
ssf  it  is  customary  to  measure  sfsf  and  divide  it  by  2.  If  it  is 
the  images  of  the  second  or  third  order  which  are  located  on  the 
scale  instead  of  those  of  the  first  order,  then  the  relation  evidently 
becomes  „  ,„ 

2X  =  rf  — .       or       3X  =  rf—  (5) 

The  distance  d  is  always  obtainable  from  the  maker  of  the  grating. 

160.  The  grating  spectrum.  If  the  source  sends  out,  not  mono- 
chromatic light,  but,  instead,  white  light,  the  series  of  sharply 
defined  images  of  the  source  is  found  to  be  replaced  by  a  single 
central  image  of  the  source  in  white  light  at  s,  bordered  on  either 
side  by  broad  bands  of  colored  light.  In  the  first  band  the  end 
farther  from  the  source  is  red.  From  the  red  the  color  grades 
into  orange,  yellow,  green,  blue-green,  blue,  and  finally  into  violet 
at  the  end  nearer  to  the  source.  The  band  of  light  thus  produced 
is  called  a  spectrum,  and  the  phenomenon  of  its  production  is 
known  as  dispersion. 

The  explanation  of  the  dispersion  produced  by  a  grating  is  as 
follows.  Just  as  in  sound  the  peculiar  tone  quality  of  any  note 
depends  upon  the  combinations  of  wave  lengths  (i.e.  the  various 
overtones)  which  enter  into  its  composition,  so  in  light  the  tint  or 
color  quality  of  a  light  depends  upon  the  wave  lengths  of  the  light 
vibrations  which  compose  it.  Thus  pure  white  light  contains  vibra- 
tions of  all  the  wave  lengths  which  are  capable  of  exciting  the 
nerves  of  the  eye.  Color  is  then  entirely  dependent  upon  wave 


THE   DIFFRACTION  GRATING  263 

length  and  corresponds  to  pitch  in  sound.  There  are  in  fact  as 
many  pure  colors  as  there  are  wave  lengths  for  visible  vibrations, 
and,  in  addition,  there  are  an  infinite  number  of  combinations  of 
color,  just  as  there  are  of  combinations  of  tones. 

The  action  of  the  grating  in  producing  dispersion  is  then  easily 
seen ;  for  since  the  position  of  every  image  except  the  central 
one  is  determined  by  the  condition  sin  6  =  n\/dy  it  is  evident  that 
there  is  a  different  value  of  6  corresponding  to  each  value  of  X. 
Now  the  wave  lengths  which  compose  white  light  vary  from  about 
.000076  in  the  red  to  about  .000039  in  the  violet.  Hence  when 
the  source  is  white  light  the  image  of  each  order  as  it  appears 
with  monochromatic  light  is  replaced  by  a  series  of  adjacent  images 
in  different  colors,  each  image  corresponding  to  a  particular  wave 
length  or  color.  This  series  of  adjacent  images  constitutes  the 
colored  band  or  spectrum  of  each  particular  order.  The  central 
image  is  white  and  sharply  defined  because  the  wave  front  AB 
(Fig.  182)  which  gives  rise  to  this  image  is  at  the  same  distance 
from  each  of  the  openings,  and  in  consequence  this  wave  front  is 
the  same  for  all  wave  lengths. 

The  spectrum  of  the  first  order  is  the  only  pure  spectrum  which 
a  grating  can  produce,  for  it  can  be  shown  that  the  spectra  of 
higher  orders  overlap.  Thus,  since  for  the  red  of  the  second  order 
sin  6  =  2  x  .00007/6?,  approximately,  and  since  for  the  violet  of 
the  third  order  sin  0'=  3  x  .00004/d,  it  will  be  seen  that  sin  9  is 
greater  than  sin  6',  and  hence  that  a  part  of  the  third  violet  over- 
laps a  part  of  the  second  red.  It  is  on  account  of  this  overlapping 
that  one  never  sees  more  than  two  or  three  spectra  on  a  side,  for 
in  the  higher  orders  the  overlapping  is  so  complete  as  to  reproduce 
white  light. 

A  grating  spectrum  is  called  a  normal  spectrum,  because  the 
distance  of  each  color  from  the  central  image  is  directly  propor- 
tional to  its  wave  length,  so  long  as  6  is  small. 

161.  The  dispersive  power  of  a  grating.  It  will  be  evident 
at  once  from  Figure  182  that  the  smaller  the  distance  between 
openings,  the  farther  apart  will  be  the  successive  images  a,  c,  d ; 
in  other  words,  that  the  angular  separation  of  different  orders 
produced  by  a  grating,  and  hence,  also,  the  angular  separation 


264  ELECTRICITY,  SOUND,  AND  LIGHT 

of  different  colors  in  the  same  order,  increases  as  the  distance 
between  the  lines  of  the  grating  decreases.  This  may  be  seen 
even  more  clearly  by  considering  the  equation  sin  9  =  n\/d  ;  for 
if  we  subtract  from  this  equation,  as  applied  to  one  color  of 
wave  length  \,  namely  sin  6l  =  n\/d,  the  same  equation  as  it 
appears  when  it  is  applied  to  another  color  of  wave  length  X2, 
namely  sin02  =  n\/d,  we  obtain 


If  we  consider  that  the  angles  6l  and  02,  being  in  general  small, 
are  approximately  proportional  to  their  sines,  we  see  that  the 
angular  separation  6^  —  #2  of  any  two  colors  Xx  and  X2  is  inversely 
proportional  to  the  grating  space  d  and  directly  proportional  to 
the  order  n  of  the  spectrum.  This  last  result  means,  of  course, 
that  the  spreading  apart  of  the  colors  is  twice  as  great  in  the 
second  spectrum  as  in  the  first,  three  times  as  great  in  the  third 
as  in  the  first,  etc. 

162.  The  resolving  power  of  a  grating.  When,  however,  we 
consider  the  sharpness  with  which  the  outline  of  any  particular 
image  in  a  particular  color  is  formed,  we  find  that  this  depends, 
for  a  given  order  of  spectrum,  not  at  all  upon  the  closeness  of 
the  lines,  but  solely  upon  the  number  of  lines  constituting  the 
grating.  Thus,  if  there  are  but  few  openings,  the  individual 
images,  a,  c,  d  (Fig.  182),  are  found  to  be  very  indistinct  in  out- 
line, but  if  there  are  a  large  number  of  openings  these  images 
become  very  sharply  defined.  In  order  to  appreciate  the  reason 
for  this  let  us  consider  a  plane  drawn  through  m(Fig.  182),  so  as 
to  make  a  very  slight  angle  with  mD,  for  example  so  that  po, 
instead  of  being  exactly  equal  to  X,  is  just  a  trifle  less  than  X, 
say  -j9Qp^  X.  The  disturbances  which  at  any  given  instant  have 
reached  this  plane  will  an  instant  later  have  passed  through  the 
lens  and  been  brought  together  to  the  point  c"  on  the  perpen- 
dicular to  this  plane  through  C.  Now,  although  the  disturbance 
which  comes  from  p  differs  but  very  slightly  in  phase  from  that 
which  comes  from  m,  the  disturbance  from  s  differs  twice  as  much 
from  that  which  comes  from  m,  etc.,  so  that  the  disturbance  from 
the  fiftieth  opening  is  just  one  half  wave  length  behind  that  from 


THE  DIFFRACTION  GRATING  265 

m,  and  therefore  completely  neutralizes  it  at  c".  If,  then,  there  are 
100  openings,  there  will  be  complete  interference  at  c",  although  c" 
is  a  point  whose  distance  from  c  is  but  yl^  of  ac.  Just  below  c" 
there  will  be  a  succession  of  maxima  and  minima  in  precisely 
the  same  way  in  which,  there  are  maxima  and  minima  about  af 
as  explained  on  page  249 ;  but,  as  has  been  already  shown,  these 
maxima  will  decrease  rapidly  in  intensity,  so  that  we  may  say 
that  practically  there  is  complete  interference  between  c"  and  a 
point  as  far  above  a  as  c"  is  below  c.  If  there  are  1000  open- 
ings, then  the  first  point  below  c  at  which  complete  interference 
occurs,  will  be  but  y-oVo  °^  ^e  distance  ac,  instead  of  yl-^  of  this 
distance.  Thus  the  larger  the  number  of  openings  the  closer  does 
the  first  point  of  complete  interference  approach  to  c,  and  there- 
fore the  more  nearly  does  the  grating  become  able  to  produce  at 
c  a  point  image  of  a  point  source ;  i.e.  in  technical  terms,  the 
higher  becomes  the  resolving  power  of  the  grating. 

If  to  the  successive  orders  c,  d,  etc.,  of  a  given  grating  (Fig.  182) 
we  apply  precisely  the  same  reasoning  as  that  given  above,  we 
see  that,  since  the  difference  in  path  po  corresponding  to  the  image 
d  is  2  X  instead  of  X,  the  first  point  of  complete  interference  near 
to  d  is  but  half  as  far  from  d  as  is  the  case  with  the  correspond- 
ing point  in  the  neighborhood  of  c.  Hence  we  see  that  with  a 
given  grating  the  resolving  power  is  proportional  to  the  order  of 
the  spectrum  observed.  All  of  the  efforts,  therefore,  which  have 
been  made  toward  increasing  the  resolving  power  of  gratings 
have  been  directed  either  toward  increasing  the  number  of  lines, 
or  else  toward  making  it  possible  to  work  in  spectra  of  a  very 
high  order. 

The  object  of  producing  gratings  of  a  very  high  resolving  power 
is,  in  general,  to  obtain  spectra  of  very  great  purity,  so  that  colors 
of  but  the  slightest  difference  in  wave  length  may  yet  stand  out 
as  distinct  colors,  that  is,  as  separate  spectral  lines.  The  largest 
and  most  perfect  gratings  thus  far  made,  namely  those  ruled  by 
Professor  Michelson  in  1908  at  the  University  of  Chicago,  contain 
nine  inches  of  ruled  surface,  the  length  of  the  lines  being  four  and 
a  half  inches,  and  the  number  of  lines  to  the  inch  12,700.  The 
total  number  of  lines  on  these  gratings  is  therefore  114,300. 


266 


ELECTRICITY,  SOUND,  AND  LIGHT 


163.  Effects  of  finite  width  of  slit  on  series  of  images  produced 
in  monochromatic  light.    In  the  use  of  a  grating  one  often  observes 
that  the  image  of  the  third  order,  for  example,  will  be  brighter 
than  that  of  the  second,  that  of  the  fifth  brighter  than  that  of  the 
fourth,  etc.    The  cause  of  this  lies  in  the  finite  width  of  the  open 
spaces  which  have  heretofore  been  considered  to  be  mere  lines. 
In  order  to  understand  the  effect  of  a  finite 
width  in  the  openings  upon  the  relative  bright- 
ness of  the  successive  images,  consider  Figure  184, 
in  which  is  shown  on  a  large  scale  a  section  of  a 
portion  of  a  practical  grating,  mo,  pr,  and  su  rep- 
resenting the  finite  openings,  and  op,  rs,  etc.,  the 
opaque  spaces.    The  points  m,  p,  and  s  correspond 
to  the  line  openings  of  the  preceding  discussion 
(Fig.  182).     The   line  mD  represents   the  wave 
front  which  gives  the  image  of  the  first  order. 
It   is    drawn   as    the  envelope   to  the   spherical 
waves  due  to  the  particles   m,  p,  and  s.     The 
disturbances  produced  by  these  particles  at  the 
points  m,  li,  and  k  are  all  in  the  same  phase  of 
vibration,  for  each  point  differs  from  the  next  in 
0     its  distance  from  its  respective  slit  by  a  whole  wave 
length.    The  disturbances  due  to  these  points  will 
then  reenforce  each  other  at  the  image  c  formed 
by  the  lens  (Fig.  182).     Similarly,  particles  such 
as  n,  q,  and  t,  which  bear  the  same  relation  in  position  to  m, 
p,  and  s  respectively,  will  also  produce  disturbances  on  the  line 
mD  at  points  which  differ  successively  in  their  distances  from 
these  particles  by  a  whole  wave  length.    The  disturbances  due  to 
these  points  will  therefore  reenforce  each  other  at  c.    And  similarly, 
for  all  the  other  points  in  the  opening  mo  there  will  be  corre- 
sponding points  in  the  other  openings  which  will  reenforce  these 
vibrations.    But  it  is  now  to  be  noticed  that  the  disturbances  winch 
start  from  the  different  parts  m,  n,  and  o  of  the  same  opening  are 
not  in  quite  the  same  phase  of  vibration  when  they  reach  the  plane 
mD,  and  further  that  the  wider  the  openings  the  greater  becomes 
this  phase  difference.     Suppose,  then,  that  the  open  spaces  mo, 


IMG.  184 


THE  DIFFRACTION  GRATING  267 

etc.,  are  just  equal  to  the  opaque  spaces  op,  etc.,  and  that  we  are 
considering  the  image  of  the  second  order.  We  have  seen  that 
the  condition  which  must  hold  for  this  image  is  that  ph  =  2\, 
sk  —  ph  =  2  X,  etc.  Since,  then,  mo  =  op,  we  have  of  =  X.  But 
when  this  condition  exists  the  disturbance  from  n  is  one  half  wave 
length  behind  that  from  m,  and  therefore  completely  destroys  it 
at  c.  Similarly,  the  disturbances  from  all  the  points  between  n  and 
o  destroy  at  c  the  disturbances  from  the  points  between  m  and  n.* 
Similarly  for  all  the  other  openings.  Thus  the  image  of  the  second 
order  will  be  entirely  missing,  and  also,  for  exactly  the  same  reason, 
the  images  of  the  fourth,  sixth,  etc.,  orders.  If  the  opening  is  one 
third  of  the  grating  space  instead  of  one  half,  the  missing  images 
will  be  those  of  the  third,  sixth,  ninth,  etc.,  orders.  If  conditions  of 
this  sort  are  only  approximately  fulfilled,  as  is  usually  the  case,  the 
images  considered  will  be  simply  weakened  but  not  entirely  cut  out. 

164.  Reflection  gratings.  By  far  the  greater  part  of  spectro- 
scopic  work  is  now  done  with  the  aid  of  gratings,  but  in  actual 
work  reflection  gratings  are  much  more  common  than  transmission 
gratings  like  that  which  has  been  studied.  These  reflection  grat- 
ings are  made  by  ruling  very  fine  lines  on  a  reflecting  metal  sur- 
face, rather  than  on  a  transmitting  glass  surface.  The  grooves 
destroy  the  light,  while  the  spaces  between  them  reflect  it  regu- 
larly. The  light  from  any  white  source  which  is  reflected  from 
such  a  grating  and  then  brought  to  a  focus  by  means  of  a  lens 
shows  a  central  white  image  at  a  position  such  that  the  angle  of 
incidence  equals  the  angle  of  reflection  (see  sect.  155).  On  either 
side  of  this  central  image  are  found  spectra  of  the  first,  second, 
and  third  orders,  precisely  as  in  the  transmission  grating.  The 
theory  of  the  two  gratings  is  in  all  respects  identical,  for  it  obvi- 
ously makes  no  difference  how  the  lines  m,  p,  s,  etc.,  of  Figure  182 
become  sources  of  disturbance,  whether  by  reflecting  or  by  trans- 
mitting a  disturbance  from  some  other  source. 

A  form  of  grating  which  has  rendered  possible  some  of  the  most 
important  of  recent  advances  in  spectroscopy  is  the  concave  grat- 
ing invented  by  the  late  Professor  Henry  A.  Rowland  of  Johns 

*  See  section  154,  page  242. 


268  ELECTRICITY,  SOUND,  AND  LIGHT 

Hopkins  University.  The  essential  difference  between  this  and 
other  gratings  is  that  the  lines  are  ruled  upon  the  surface  of  a 
concave  spherical  mirror  of  large  radius  of  curvature,  for  example 
20  feet.  Under  such  conditions  the  mirror  itself  forms  the  series 
of  images  corresponding  to  a,  c,  d,  etc.  (see  Fig.  182,  and  also  Fig. 
180),  so  that  it  is  not  necessary  to  interpose  a  lens.  This  eliminates 
all  difficulties  arising  from  the  absorption  of  the  waves  by  the  lens, 
difficulties  which  are  especially  pronounced  in  the  ultra-violet  and 
infra-red  regions  of  the  spectrum. 

EXPERIMENT  22 

Object.  To  determine  the  wave  length  of  sodium  light  by  means  of  a 
diffraction  grating. 

Directions.  Over  a  Bunsen  burner  place  a  sheet-iron  chimney  C  (Fig.  185) 
in  which  is  cut  a  narrow  vertical  slit  .<?,  triangular  in  form.  Immediately 
behind  this  place  a  horizontal  scale  at  the  same  height  as  the  slit.  Set  up 
a  transmission  grating  in  a  clamp  three  or  more  meters  away  from  the  scale, 
and  adjust  its  position  until  it  is  parallel  to  the  scale  and  lies  on  the  per- 


FIG.  185 


pendicular  drawn  to  the  scale  through  the  slit.    Exclude  the  £ 

air  from  the  burner  so  that  it  burns  with  a  white  flame,  and 
view  the  slit  through  the  grating.    We  then  have  the  conditions  described 
in  section  160.    Note  the  overlapping  of  colors  in  spectra  of  higher  order 
than  the  first. 

Replace  the  white  light  by  sodium  light  by  readmitting  the  air  to  the 
burner  and  placing  a  bit  of  asbestos  soaked  in  salt  water  in  the  colorless 
flame.  Keep  the  eye  very  close  to  the  grating  in  order  that  distances  like 
s'E  may  be  measured  to  the  grating  rather,  than  to  the  eye  without  intro- 
ducing an  appreciable  error.  Slide  a  narrow  and  pointed  paper  rider  r  over 
the  scale  and  locate  with  it  on  the  scale  the  positions  of  the  images  ,<  s'^ 
/,'  s",  and  so  on.  Take  the  grating  space  d  either  from  the  data  furnished 
by  the  maker  of  the  grating  or  from  a  direct  determination  with  a  microm- 
eter microscope.  Measure  the  distances  Es',  Es{,  and  so  on,  with  a  steel- 


THE  DIFFRACTION  GRATING  269 

tape.  From  the  mean  of  Es'  and  Es\,  and  from  one  half  of  s'.s^,  calculate 
X  by  the  use  of  the  relations  developed  in  section  159.  Compute  X  also 
from  measurements  on  the  brightest  image  of  the  higher  orders  observed. 

Obtain  X  from  a  similar  set  of  observations  upon  the  brightest  image 
formed  by  a  grating  which  has  a  different  value  of  d. 

Compare  the  mean  of  all  the  results  with  the  accepted  value  for  sodium 
light,  namely  .00005.89  cm. 

EXAMPLE 


Using  a  grating  with  a  space  d  of  .002  cm.,  the  distance  s's{  was  found 
be  12.35  cm.     T] 
respectively.    Hence 


to  be  12.35  cm.     The  distances  Es'  and  Es{  were  209.2  and  209.8  cm. 


(6.17)  (.002) 
(2,09.5) 

The  images  of  the  fourth  order  were  bright  and  easily  located.    For  these 
s""  g^"  was  found  to  be  49.7,  Es""  =  209.6,  and  Es'{"  =  212.3.    Hence 

(24. 85)  (.002) 

X  =  - —^— — '-  =  .00005890. 

4(210.95) 

Using  a  second  grating  for  which  d  was  .001,  the  images  of  the  second 
order  were  selected  for  the  calculation  ;  s"x'{  was  24.75,  Ea"  was  209.2,  and 
Es"  was  210. 3.  Hence  X  was  .00005900.  These  results  all  agreed  to  within 
.2%  with  the  value  .00005890. 


CHAPTEE   XXIII 


THE  REFRACTION  OF  LIGHT 

165.  Cause  of  refraction.  It  is  the  change  in  the  wave  front 
which  occurs  when  a  wave  enters  a  new  medium  that  accounts 
for  the  bending  which  light  undergoes  when  it  passes,  at  any 
other  than  normal  incidence,  from  one  medium  into  another. 
Thus,  if  a  wave  which  originates  at  p  (Fig.  186)  has  its  form  so 
changed  by  passage  into  a  new  medium  that  the  center  of  the 

wave  front  which  reaches  the 
eye  at  E  is  at  p'  instead  of 
at  p,  then  obviously  the  light 
which  has  come  to  E  from 
p  has  come  over  the  broken 
path  pqJS.  That  is,  light  which 
passes  obliquely  from  a  medium 
of  slower  speed  to  one  of  higher 
speed  is  bent  away  from  the 
perpendicular  qn  drawn  into 
the  second  medium. 

If  the  speed  in  the  second 
medium  had  been  less  than  in 
the  first,  then  obviously  the 
new  center  p'  would  have  been 
below  p  instead  of  above  it. 
This  means  that  light  when  passing  from  a  medium  of  greater  to 
one  of  lesser  speed  is  lent  toward  the  perpendicular  drawn  into  the 
second  medium. 

This  change  in  direction,  due  to  change  in  wave  form,  which 
light  undergoes  in  passing  from  one  medium  to  another  is  known 
as  refraction.  In  order  to  be  in  position  to  make  a  quantitative 
study  of  the  phenomena  of  refraction  it  is  first  desirable  to  obtain 
a  quantitative  expression  for  the  curvature  of  a  wave  front. 

270 


THE   EEFKACTION   OF  LIGHT 


271 


166.  Measure  of  curvature.  A  circle  has  obviously  the  same 
curvature  at  every  point.  In  Figure  187  are  shown  two  circles, 
one  of  twice  the  radius  of  the  other.  It  is  evident  from  the  figure 
that  in  moving  along  the  larger  circle  from  0  to  a  the  curve 
departs  from  the  tangent  (that  is,  curves)  much  less  than  it  does 
in  moving  an  equal  distance  from  0  to  a'  along  the  smaller  circle ; 
that  is,  the  greater  curvature  accompanies  the  smaller  radius. 
It  is  further  evident  from  the 
figure  that  as  Oa  and  Oaf  be- 
come smaller  and  smaller,  the 
distance  from  a  to  the  tangent 
becomes  more  and  more  nearly 
exactly  one  half  of  the  distance 
from  a'  to^-the  tangent.  At  0 
therefore  the  rate  at  which  the 
smaller  circle  is  curving  away 
from  the  tangent  line  is  just 
twice  the  rate  at  which  the 
larger  circle  is  curving  away 
from  this  tangent.  In  other 

words,  the  curvatures  of  two  circles  are  inversely  proportional  to 
their  radii.  Hence  the  reciprocal  of  the  radius  of  any  circle  is 
taken  as  the  measure  of  the  curvature  of  that  circle.  Thus  if  C 
denote  the  curvature  of  any  curve  at  any  point,  and  R  the  radius 
of  curvature  of  the  curve  at  this  point,  we  have  by  definition 


c-1 


(1) 


167.  Ratio  of  the  velocities  of  light  in  two  media.    If  it  may 

be  regarded  as  established  by  the  experiments  which  have  preceded 
that  light  is  a  wave  motion,  then  it  follows  from  the  fact  that 
objects  under  water,  for  example,  appear  nearer  to  the  surface  than 
they  actually  are,  that  light  travels  faster  in  air  than  it  does  in 
water,  for  it  is  only  in  view  of  an  increase  in  speed  in  going  from 
water  to  air  that  we  can  account  for  such  an  upward  bulging  of 
the  emerging  wave  as  would  bring  nearer  to  the  eye  the  center 
from  which  the  waves  appear  to  proceed  (see  Fig.  186). 


72 


ELECTRICITY,  SOUND,  AND  LIGHT 


Water 


FIG.  188 


But  this  phenomenon  of  the  apparent  elevation  of  objects 
under  water  does  more  than  merely  to  establish  the  fact  of  a 
difference  in  the  speed  of  light  in  the  two  media :  it  enables  us  to 

determine  fairly  accurately 
the  ratio  of  the  two  speeds. 
For  let  cod  (Fig.  188),  drawn 
with  S  as  a  center,  represent 
a  section  of  the  wave  front 
at  the  instant  it  touches  the 
water  surface  ww'  at  o ;  let 
mo'n  represent  a  section  of 
the  wave  front  as  it  would 
have  been  an  instant  later 
if  there  had  been  no  change 
in  medium,  and  let  mo"n, 
drawn  with  S'  as  a  center, 
represent  the  wave  front  as 
it  actually  is  at  this  instant. 
Then  obviously  the  speed  of  light  in  air  divided  by  its  speed  in 
water  is  oo"/oo'.  But  oo"  and  oo'  measure  respectively  the  curva- 
tures of  the  arcs  mo"n  and  mo'n,  so  long  as  these  arcs 
are  small,  for  then  they  are  the  amounts  by  which 
these  curved  lines  depart  from  the  straight  line  ww' . 
Hence,  since  curvatures  are  also  measured  by  the 
reciprocals  of  the  radii  of  curvature  (eq.  1),  we  have 

velocity  of  light  in  air     _  oo"  _  Sm 
velocity  of  light  in  water      oo'       S'm 

Now  as  one  looks  down  vertically  upon  a  surface 
of  water,  the  section  of  the  wave  taken  in  by  the 
two  eyes  is  so  small  that  Sm  and  S'm  are  practi- 
cally the  distances  from  S  and  Sr  respectively  to 
the  surface.  Hence,  to  measure  the  ratio  of  the 
velocities  of  light  in  air  and  water  we  have  only 
to  look  down  into  a  tall  vessel  of  water,  as  in  Figure  189,  place 
a  finger  on  the  outside  of  the  vessel  at  the  apparent  level  of  the 
bottom,  and  divide  the  actual  depth  by  this  apparent  depth. 


FIG.  189 


THE  KEFKACTION  OF  LIGHT  273 

With  the  aid  of  a  microscope  this  method  is  often  extended  to 
the  determination  of  the. ratio  of  the  speed  of  light  in  air  and  in  any 
transparent  medium  which  is  bounded  by  parallel  planes.  Thus 
the  microscope  is  first  focused  upon  a  point  p  (Fig.  190)  when 
air  only  intervenes  between  p 
and  the  objective  0  of  the  micro- 
scope. Then  the  medium  with 
the  parallel  faces  ab  and  cd  is  in- 
troduced between  p  and  0.  This  r 
causes  the  center  of  the  wave 

which  reaches  0  to  change  from    _ 

p  to  some  point  p'.    Hence  in 
order  to  see  p  distinctly  it  is  now 

necessary  to  raise  the  microscope  a  distance  pp1.  The  amount  of 
this  elevation  is  carefully  measured.  Then  the  microscope  is  again 
raised  until  the  surface  ab  is  in  focus,  i.e.  it  is  raised  the  dis- 
tance p'n.  The  ratio  of  the  velocities  of  light  in  air  and  in  the 
given  medium  is  then  pn/pfn. 

168.  Elementary  lens  and  mirror  formulas.  In  section  155  we 
have  already  discussed  qualitatively  the  refraction  produced  by 
lenses.  In  the  light  of  section  166  it  now  becomes  very  easy  to 
deduce  the  formula  by  means  of  which  the  distance  f  (i.e.  Lp, 
Fig.  175)  of  the  center  of  a  transmitted  wave  from  the  lens  is  given 
in  terms  of  the  focal  length  F  of  the  lens  and  the  distance  f 
(i.e.  SL,  Fig.  175)  of  the  source  from  the  lens.  For,  since  the  focal 
length  F  is  defined  as  the  distance  from  the  lens  of  the  center  of 
a  transmitted  wave  which  before  incidence  was  plane,  i.e.  had  zero 
curvature,  it  follows  from  the  definition  of  curvature  (eq.  1)  that 
the  curvature  which .  a  lens  imparts  to  a  wave  passing  through  it 
is  1/F.  It  follows  further  from  this  definition,  and  from  the  fact 
that  a  convex  lens  always  tends  to  reverse  the  direction  of  cur- 
vature of  a  wave  approaching  it  from  any  source  (see  sect.  155),  that 
the  curvature  I//'  of  any  transmitted  wave  must  be  the  difference 
between  the  curvature  I//  of  the  incident  wave  and  the  curvature 
l/F  imparted  by  the  lens.  That  is, 

111  111 


274 


ELECTRICITY,  SOUND,  AND  LIGHT 


This  formula  applies  also  to  concave  mirrors,  since  these  also 
tend  to  reverse  the  curvature  of  an  incident  wave  (see  Figs.  173 
and  174,  pp.  248-249).  The  formula  shows  at  once  that  the  focal 
length  F  of  a  mirror  is  half  its  radius  of  curvature  R  ;  for  when  a 
source  S  (Fig.  174)  is  at  the  center  of  curvature  of  a  concave  mirror 
we  have  /  =/'  =  R.  Hence  from  (2) 

l  =  f,     or 


-  a* 


Since,  as  constructions  similar  to  those  given  in  section  155 
will  show,  concave  lenses  and  convex  mirrors  tend  to  increase  the 
curvature  of  incident  waves,  the  corresponding  formula  which 
holds  for  either  of  these  is  obviously 

111  111 

7~^+7'          f   J-F 

169.  Index  of  refraction.  By  considering  a  very  small  section 
of  the  wave  front  at  q  (Fig.  191)  it  is  easy  to  find  a  relation 
between  the  amount  of  bending  which  light  undergoes  in  passing 
from  one  medium  to  another  and  the  speeds  of  propagation  of 

light  in  the  two  media.  Thus  let 
qt  represent  so  minute  an  element 
of  the  wave  front  at  q  that  it  may 
be  considered  as  plane.  For  con- 
venience of  representation  let  it  be 
greatly  magnified,  as  in  the  figure. 
When  the  left  edge  of  this  element 
meets  the  new  medium  at  q,  this 
point  becomes  the  center  of  a  new 
disturbance  which  spreads  as  a  cir- 
cular wave  into  the  new  medium. 
If  the  speed  of  light  in  this  medium 
is  greater  than  is  its  speed  in  the 
lower  medium,  then  while  the  disturbance  from  t  travels  a  dis- 
tance tv  in  the  lowe.r  medium,  that  from  q  travels  a  greater  dis- 
tance qii  in  the  upper  medium.  The  disturbances,  then,  which  at 
a  given  instant  started  from  the  various  points  along  the  line  qt 
will  an  instant  later  have  reached  points  which  lie  along  the  line 


THE  KEFRACTION  OF  LIGHT 


275 


uv,  that  is,  uv  is  the  new  wave  front  in  the  upper  medium.  Whereas, 
then,  the  light  approached  the  interface  between  the  media  travel- 
ing in  the  direction  oq,  it  recedes  from  it  traveling  in  the  direction  qp. 
Now  if  we  designate  by  i  the  angle  nqp  which  the  direction  of 
propagation  of  the  light  in  the  medium  of  greater  speed  makes 
with  the  normal  drawn  into  this  medium,  and  by  r  the  angle  oqn' 
which  the  direction  of  propagation  of  the  light  in  the  medium 
of  lesser  speed  makes  with  the  normal  drawn  into  this  medium, 
and  if  we  designate  by  Vi  the  speed  of  propagation  in  the  former 
medium  and  by  Vr  that  in  the  latter  medium,  then  we  have 

rp,  -  OU  Sllli 

Therefore  --  =  — 

to        sin?' 


£  =  *=•     But 

Vr       tv 

Hence 


=  su 


A     tV 

and  —  =  sin 


_! 

Vr 


sn  r 


(5) 


It  is  customary  to  call  this  ratio  the  relative  index  of  refraction 
of  the  two  media.  If  the  upper  medium  is  a  vacuum,  then  this 
ratio  is  called  the  absolute  index  of  refraction  of  the  lower  medium, 
and  is  denoted  by  the  letter  n.  The  speed  of  propagation  in  air  is 
so  nearly  that  in  a  vacuum  that  for  most  purposes  it  is  sufficiently 
accurate  to  consider  the  index  of  refraction  of  any  medium  as  the 
ratio  of  the  velocities  of  light  in  air  and  in  the  medium.  We  have, 
then, 


sin 


170.  Refraction 
through  a  prism. 
Consider  a  prism 
composed  of  some 
refracting  substance 
such  as  glass  (Fig. 
192)  and  surrounded 
by  air.  A  beam  of 

light  OP  is  incident  upon  one  of  its  faces  at  an  angle  i,  and  is 
refracted  at  an  angle  of  r.  This  refracted  beam,  incident  upon  the 
second  face  at  an  angle  r',  undergoes  further  deviation,  emerging 


276 


ELECTRICITY,  SOUND,  AND  LIGHT 


into  the  air  at  an  angle  ir.  The  angle  D  between  OP,  the  original 
direction  of  the  beam  of  light,  and  QR,  its  direction  after  passing 
through  the  prism,  represents  the  deviation  produced  in  the  direc- 
tion of  the  beam.  This  angle  is  seen  from  the  figure  to  be  express- 
ible in  terms  of  i,  r,  if,  and  rf  as  follows  : 

(i'  -  r')  =  (i  +  i')  -(r  +  /). 


If  A  is  the  angle  of  the  prism,  we  have,  since  r  +  rf=A  (see  sect.  172), 


171.  Angle  of  minimum  deviation.  It  is  evident  that  if  the 
beam  of  light  in  Figure  192  is  caused  to  enter  the  prism  in  the 
reverse  direction,  namely  RQ,  it  will  travel  over  exactly  the  same 
path  and  suffer  the  same  deviation.  That  is,  there  are  two  values 
of  the  angle  of  incidence  of  the  entering  beam  for  which  the  devi- 
ation is  the  same,  namely  i  and  ir.  Consequently,  if  the  angle  of 
incidence  be  caused  to  vary  from  a  value  i  to  a  value  i',  it  follows, 
since  the  deviation  D  changes  continuously  between  these  values, 
and  has  the  same  value  for  an  incidence  of  i  as  for  one  of  ir,  that 
it  must  pass  through  a  maximum  or  a  minimum.  A  simple  experi- 
ment to  be  described  later  shows  that  it  passes  through  a  minimum. 
Further,  since  this  minimum  value  of  D  must  occur  for  an  angle 
of  incidence  which  lies  between  i  and  i',  no  matter  how  slightly 
i  and  i1  differ,  it  follows  that  it  must  occur  when  i  equals  i'.  Hence 

when  the  deviation  is 
a  minimum  we  have 

D=2i-A.      (8) 

172.  Index  of  re- 
fraction from  meas- 
urements upon  the 
angle  A  of  a  prism 
and  the  angle  of  min- 

imum    deviation   D. 

FIG.  193  The  index  of  refrac- 

Q-j-r-x    n 

tion  lias  been   shown   to  be  given  by  the  expression  n  =  — — . 
For  the  condition  of  minimum  deviation  as  given  in  equation  (8), 


THE  REFRACTION   OF  LIGHT 


277 


we  have  i  =  (A  +  I>)/2.  To  obtain  r  for  this  case,  consider  Figure 
193,  from  which  it  is  evident  that  <f>  +  A  =  lSQ°  and  </>  +  2  r  =  180° ; 
hence  r  =A/2.  Substituting  these  values  in  the  expression  for  the 
index  of  refraction,  we  obtain 

sin  J  (A  +  D) 


sin  J-  A 


(9) 


173.  The  spectrometer.  The  spectrometer  is  an  instrument  of 
the  form  shown  in  Figure  194.  Its  essential  features  are  repre- 
sented diagrammatically  in  Figure  195.  A  circular  table  K,  the 


FIG.  194 

edge  of  which  is  graduated  in  degrees,  is  supported  upon  a  mount- 
ing which  carries  also  a  telescope  T  and  a  so-called  collimator  C. 
The  latter  consists  merely  of  a  tube  carrying  a  slit  s  so  mounted 
that  it  may  be  placed  in  the  principal  focal  plane  of  a  lens  L'. 
The  object  of  this  arrangement  is  to  make  it  possible  to  regard  s 
as  an  infinitely  distant  source  of  light,  for  waves  which  originate 
at  s  become  plane  waves  after  passing  through  the  lens  L'.  The 
telescope  T  is  mounted  so  as  to  rotate  about  the  axis  of  the  table 
K.  The  angular  position  of  the  telescope  with  reference  to  the 


278 


ELECTRICITY,  SOUND,  AND  LIGHT 


graduations  on  the  table  may  be  read  with  the  aid  of  a  vernier 
V  attached  to  the  telescope.  Attached  to  the  circular  table  is  a 
second  smaller  circular  plate  called  the  prism  table,  which  may  be 

leveled  by  means  of  the  level- 
ing screws  E  (see  also  Fig.  198). 
This  table  carries  the  prism  P. 
The  telescope  may  be  clamped 
to  the  mounting,  and  the  cir- 
cular table,  with  the  attached  prism 
table,  rotated.  The  rotation  with  ref- 
erence to  the  fixed  telescope  may 
then  be  read  with  the  aid  of  the 
vernier  V.  A  small  piece  of  plane 
transparent  glass  m  is  inserted  in 
the  eyepiece  e  so  as  to  make  an 
angle  of  45°  with  the  axis  of  the 
telescope.  The  purpose  of  this  ar- 
rangement is  to  make  it  possible  to 
illuminate  the  cross  hairs  at  x  by 
throwing  a  beam  of  light  from  a 
flame  or  other  source  /  into  the 
eyepiece  through  the  circular  open- 
ing 0,  and  thence,  after  reflection 
from  the  surface  of  m,  down  the  axis  of  the  telescope  tube.  An 
eyepiece  provided  with  the  opening  0  and  the  glass  plate  m  is 
called  a  Gauss  eyepiece. 

174.  Measurement  of  the  angle  of  a  prism.  The  prism  whose 
angle  A  is  to  be  measured  is  placed  upon  the  prism  table  as  shown 
in  Figure  195.  Now  if  the  illuminated  cross  hairs  x  are  in  the 
focal  plane  of  the  lens  L,  then  the  light  which  passes  from  them 
down  the  tube  may  be  reflected  from  the  prism  face  P  so  as  to 
return  into  the  telescope,  in  which  case  it  will  be  brought  again  to 
a  focus  in  the  focal  plane  of  the  lens  L.  Hence  an  eye  looking  into 
the  eyepiece  should  see  side  by  side  two  images  of  the  cross  hairs, 
one  due  to  light  which  comes  directly  from  the  cross  hairs  at  x  to 
the  eye,  the  other  due  to  light  which  has  passed  down  the  tube  and 
been  reflected  back  again  into  the  tube  from  the  face  P.  If  the 


FIG.  195 


THE  REFRACTION  OF  LIGHT 


279 


prism  table  is  turned  until  these  two  images  of  the  cross  hairs  are 
brought  into  exact  coincidence,  we  may  know  that  the  axis  of  the 
telescope  is  strictly  perpendicular  to  the  face  P.  If  now  the  prism 
table  is  rotated  until  a  reflected  image  of  the  cross  hairs  is  obtained 
from  the  face  P',  and  if  this  image  is  brought  into  coincidence 
with  the  direct  image,  the  axis  of  the  telescope  will  be  perpen- 
dicular to  P'.  The  angle  through  which  the  prism  has  been  turned, 
as  read  upon  the  vernier  and  scale,  is  the  supplement  of  the  angle 
of  the  prism,  that  is,  it  is  180°  —  A. 

The  second  method  of  measuring  the  angle  of  a  prism  is  one 
which  makes  use  of  the  collimator.  Suppose  that  the  slit  s  has 
been  placed  in  the  focal  plane  of  the  lens  L1 ',  and  that  the  cross 
hairs  x  have  been  placed  « 

in  the  focal  plane  of  the 
lens  L.  Then,  when  the 
light  from  the  slit  s  is 
reflected  from  a  prism 
face  into  the  telescope  as 
in  Figure  196,  an  image 
of  s  will  be  formed  in  the 
telescope  in  the  plane  of 
the  cross  hairs.  Between 
L'  and  L  the  light  waves 
which  form  this  image 
are  plane,  or,  to  express 
the  same  idea  in  different 
terms,  the  beam  of  light 

between  L'  and  L  consists  of  a  bundle  of  parallel  rays.  If  now 
the  prism  is  set  so  that  a  part  of  this  beam  is  reflected  from  one 
face  of  the  angle  A  into  the  telescope  when  the  latter  is  in  the 
position  M,  and  another  part  is  reflected  from  the  other  face  of 
the  angle  A  into  the  telescope  when  the  latter  is  turned  into  the 
dotted  position  M',  then  the  image  of  the  slit  s  can  be  seen  in 
the  telescope  when  the  latter  is  in  either  of  these  positions.  If  the 
cross  hairs  of  the  telescope  are  set  upon  the  middle  of  this  image 
at  M,  and  the  reading  of  the  vernier  and  scale  taken,  and  if  then 
the  telescope  is  rotated  into  position  M '  and  a  similar  setting  and 


FIG.  196 


280 


ELECTRICITY,  SOUND,  AND  LIGHT 


reading  taken,  a  little  consideration  will  show  that  the  angle  A  is 
one  half  of  the  angle  through  which  the  telescope  has  been  rotated. 
This  follows  at  once  from  the  fact  that  all  of  the  lines  drawn  from 
L'  to  the  prism  are  parallel,  and  that  "  angle  of  incidence  equals 
angle  of  reflection." 

175.  Measurement  of  the  angle  of  minimum  deviation!  Since 
the  passage  of  white  light  through  a  prism  breaks  it  up  into  a 
band  of  colored  light,  the  angle  of  deviation  must  be  different  for 
different  colors.  To  obtain  the  angle  of  minimum  deviation  for  a 
particular  color,  we  first  set  the  cross  hairs  in  the  focal  plane  of  L, 
0  and  the  slit  in  the  focal  plane  of 

L',  and  then  set  up  at  o  a  source  of 
monochromatic  light,  for  example, 
a  sodium  flame.  We  next  dispose 
the  prism,  telescope,  and  collimator 
in  the  manner  shown  in  Figure  197, 
and  observe  in  the  telescope  the  im- 
age of  the  slit  formed  by  light  which 
has  undergone  refraction  in  passing 
through  the  prism.  To  make  sure 
that  the  image  seen  is  formed  by 
refraction  rather  than  by  reflection 
from  the  face  BC,  for  example,  we 
have  only  to  place  a  source  of  white 
light  at  o  and  see  whether  the  ob- 
served image  of  the  slit  is  replaced 
by  a  broad  band  of  color.  .Having 
found  the  refracted  image,  we  rotate  the  prism  table  slightly, 
observing  meantime  the  image  of  the  slit  in  the  telescope;  and 
if  this  image  moves  in  the  direction  of  decreasing  deviation, 
i.e.  toward  R,  we  follow  it  with  the  telescope  until,  without 
changing  the  direction  of  rotation  of  the  prism,  the  image  begins 
to  return  toward  L.  If  the  image  first  begins  to  move  toward  L, 
WQ  at  once  reverse  the  direction  of  rotation  and  follow  with  the 
telescope  until  the  position  of  minimum  deviation  has  been  reached, 
i.e.  until  the  beam  emerging  from  the  face  A  C  of  the  prism  makes 
the  smallest  angle  which  it  is  found  possible  for  it  to  make  with 


FIG.  197 


THE  REFRACTION  OF  LIGHT 


281 


the  direction  of  the  beam  incident  upon  the  face  AB.  We  then 
set  the  cross  hairs  upon  the  middle  of  the  slit,  the  latter  being  now 
in  this  position  of  minimum  deviation,  and  read  the  vernier  and 
scale.  We  next  remove  the  prism,  rotate  the  telescope  into  the 
position  Ny  and  set  the  cross  hairs  upon  the  image  of  the  slit 
formed  by  this  undeviated  beam.*  The  difference  between  the 
two  settings  gives  the  angle  of  minimum  deviation  D. 


EXPERIMENT  23 

Object.  To  find  the  index  of  refraction  of  glass  for  sodium  light  by  a 
measurement  (I)  of  the  angle  of  the  prism,  and  (II)  of  the  angle  of  mini- 
mum deviation. 

Directions.  I.  Set  the  prism  P  (Fig.  198)  upon  the  prism  table  in  such 
a  position  that  each  of  the  edges  is  immediately  above  one  of  the  leveling 


FIG.  108 

screws  E,  E' ',  E."  Loosen  the  set  screw  *S\,  so  that  the  prism  table  can 
rotate  by  itself.  Set  up  an  electric  light  behind  a  ground-glass  screen 
about  a  foot  away  from  the  opening  0  in  the  Gauss  eyepiece,  and  on  the 

*It  is  frequently  undesirable  to  disturb  the  adjustments  of  the  spectrometer 
by  removing  the  prism.  In  that  case  enough  light  will  generally  pass  the  edges 
of  the  prism  to  make  possible  the  setting  in  the  position  N  without  removing 
the  prism.  If  this  is  not  the  case  the  telescope  may  be  turned  into  a  position 
corresponding  with  that  already  shown  in  the  figure,  but  on  the  other  side  of  N, 
the  prism  reversed,  and  the  position  of  minimum  deviation  determined.  The 
position  N  then  lies  halfway  between  these  two  points,  or,  in  other  words,  the 
angle  through  which  the  telescope  has  turned  is  2  J). 


282  ELECTRICITY,  SOUND,  AND  LIGHT 

perpendicular  drawn  through  0  to  the  axis  of  the  telescope  (see  also  Fig. 
195).  Clamp  the  telescope  to  the  frame  by  tightening  the  set  screw  S2. 
If  the  cross  hairs  are  not  perfectly  distinct,  slip  the  eyepiece  alone  forward 
or  back  until  they  are  in  sharp  focus.  Then  rotate  the  prism  table  until 
the  light  which  enters  O  from  the  source /(Fig.  195),  and  is  then  reflected 
down  the  telescope  tube  by  the  plane  glass  m,  is  reflected  back  into  the  tube 
from  the  face  P  of  the  prism.  If  the  prism  is  being  rotated  somewhat 
rapidly,  this  reflection  will  appear  simply  as  a  flash  of  light  across  the  field 
of  view  when  the  prism  face  P  passes  through  the  position  in  which  it  is 
at  right  angles  to  the  axis  of  the  telescope.  If  this  flash  cannot  be  found 
at  first,  change  the  inclination  of  the  telescope  by  means  of  the  screw  S3 
until  it  appears.  Keeping  this  reflected  light  in  the  field  of  view,  move 
forward  or  back  the  draw  tube  which  carries  the  eyepiece  and  cross  hairs 
until  a  second  image  of  the  cross  hairs  is  seen  in  this  field  of  reflected  light.* 
The  cross-hairs  must  then  be  in  the  focal  plane  of  the  objective  lens  L 
(Fig.  195),  since  light  waves  originating  in  the  plane  of  the  cross  hairs 
return  to  a  focus  in  this  same  plane  after  passing  through  L  and  being- 
reflected  by  a  plane  surface  back  through  L.  To  make  the  adjustment 
perfect,  turn  the  prism  so  as  to  bring  the  two  images  of  the  vertical  cross 
hair  very  nearly  into  coincidence,  and  move  the  head  from  side  to  side  to 
test  for  parallax.  If  this  motion  causes  the  two  images  to  appear  to  move 
with  reference  to  each  other,  focus  again  until  such  apparent  movement 
is  eliminated.  The  telescope  is  now  said  to  be  focused  for  parallel  rays. 
This  adjustment  might  also  have  been  made  by  merely  turning  the  tele- 
scope upon  some  very  distant  object  and  focusing  until  there  was  no 
parallax  between  the  cross  hairs  and  points  on  the  object. 

To  make  the  second  adjustment,  which  consists  in  setting  the  instru- 
ment so  that  the  axis  of  the  telescope  is  strictly  perpendicular  to  the  axis 
of  rotation  of  the  prism,  turn  the  thumbscrew  Ss  until  the  two  images  of 
the  horizontal  cross  hair,  found  as  above,  coincide;  then  loosen  the  set 
screw  S4,  clamp  51?  and  rotate  the  graduated  circle,  with  the  prism,  until 
one  of  the  other  faces,  say  P'  (Fig.  198)  instead  of  P,  reflects  the  light  back 
into  the  telescope.  Unless  the  axis  of  rotation  is  already  perpendicular  to  the 
axis  of  the  telescope  the  horizontal  cross  hairs  will  now  110  longer  coincide. 
Restore  coincidence  by  taking  up  one  half  of  the  distance  between  the  two 
images  of  the  cross  hairs  with  S3,  and  the  other  half  with  the  prism  table 
screw  E  which  is  opposite  to  the  face  P'.  Then  rotate  the  graduated  circle 
until  P  is  again  perpendicular  to  the  telescope,  and  again  take  up  half  the 

*  If  the  prism  has  three  polished  faces,  there  will,  in  general,  be  three  reflected 
images  of  the  cross  hairs  instead  of  one,  two  of  them  being  due  to  double  reflec- 
tions from  the  rear  faces  of  the  prism.  To  avoid  the  confusion  arising  from 
these  double  reflections,  it  is  recommended  that  one  face  be  temporarily  painted 
with  a  mixture  of,  say,  whiting  and  alcohol,  or  lampblack  and  alcohol.  The  three 
polished  faces  are  needed  for  Experiment  24. 


THE   REFRACTION  OF  LIGHT  283 

difference  in  coincidence  by  means  of  S3  and  half  by  means  of  E" ,  the  prism 
table  screw  which  is  opposite  to  P.  After  two  or  three  repetitions  of  this 
process  the -two  images  of  the  horizontal  cross  hairs  should  coincide,  no 
matter  which  face  of  the  prism  is  turned  so  as  to  reflect  light  back  into  the 
telescope.  If  exact  coincidence  cannot  be  obtained,  it  is  because  the  edges 
of  the  prism  itself  are  not  quite  parallel.  In  this  case  it  is  customary  to 
distribute  the  error  in  coincidence  between  the  faces. 

Xext  mark  carefully  the  particular  angle  of  the  prism  which  you  wish 
to  measure ;  then  obtain  the  reflected  image  of  the  cross  hairs  from  one  of 
the  faces  of  this  angle.  See  that  the  clamp  S2  is  tight,  so  that  the  telescope 
will  henceforth  remain  quite  stationary,  rotate  the  graduated  circle  until 
the  two  sets  of  vertical  cross  hairs  are  in  approximate  coincidence,  and,  to 
make  this  adjustment  perfect,  clamp  S4  and  complete  the  coincidence  by 
means  of  the  fine-adjustment,  slow-motion  screw,  S&.  Take  the  exact 
reading,  in  degrees,  minutes,  and  seconds,  of  the  vernier  and  scale;  then 
loosen  S4,  rotate  the  graduated  circle  with  the  prism  until  the  other  face 
of  the  angle  to.  be  measured  is  perpendicular  to  the  telescope,  clamp  $4, 
make  a  final  setting,  as  before,  with  £6,  and  then  take  a  second  reading  of 
the  vernier  and  scale.  The  difference  between  the  two  readings  should  be 
the  supplement  of  the  angle  of  the  prism. 

As  a  check  upon  this  determination,  measure  the  same  angle  by  the 
second  method  of  section  174.  To  do  this,  first  see  that  the  telescope  is 
still  focused  as  above  for  parallel  rays;  then  remove  the  prism,  loosen  £„, 
and  rotate  the  telescope  so  as  to  see  through  it  into  the  collimator.  Level 
the  collimator  approximately  by  means  of  S7  until  the  cross  hairs  are  in 
the  middle  of  the  slit ;  then  loosen  the  tube  which  carries  the  slit  of  the 
collimator,  and  slip  it  in  or  out  until  the  image  of  the  slit  observed  through 
the  telescope  is  in  sharp  focus.  To  make  the  adjustment  perfect,  move  the 
slit  until  there  is  no  parallax  between  the  cross  hairs  and  the  image  of  the 
slit.  Xow  complete  the  leveling  of  the  collimator  by  adjusting  S7  again. 
Then  set  the  prism  as  shown  in  Figure  196,  clamp  $4,  loosen  S2,  and  place 
the  telescope  first  in  position  AT,  then  in  position  A/',  and  see  if  a  sharp 
reflected  image  of  the  slit  can  be  seen  in  both  positions.  If  not,  adjust  the 
position  of  the  prism  until  both  images  appear ;  then  make  the  slit  very 
narrow  by  means  of  S8,  turn  the  telescope  into  position  A/,  clamp  52,  and 
by  means  of  the  fine-adjustment  screw  S6  set  the  vertical  cross  hair  upon 
the  middle  of  the  slit,  and  then  take  the  reading  of  the  vernier  and  scale. 
Loosen  £«,,  turn  the  telescope  to  position  A/"',  and  repeat.  The  difference 
between  the  two  readings  should  be  twice  the  angle  A  of  the  prism. 

II.  To  measure  the  angle  of  minimum  deviation,  first  clamp  S4  and 
loosen  both  A^  and  S2.  Then  set  up  a  sodium  flame  at  o  (Fig.  197),  and, 
by  rotating  both  the  prism  and  the  telescope,  find  the  refracted  image  of 
the  slit  in  the  telescope  when  the  known  angle  A  is  placed  as  in  the  dia- 
gram. If  the  image  cannot  be  found  at  first,  it  is  probable  that  the  prism 


284  ELECTRICITY,  SOUND,  AND  LIGHT 

needs  to  be  rotated  somewhat  farther  in  the  direction  of  the  arrow.  Having 
found  the  refracted  image,  proceed  as  indicated  in  section  175,  setting  first 
approximately  upon  the  position  of  minimum  deviation,  then  clamping 
both  Sl  and  S2,  and  setting  the  cross  hairs  exactly  on  the  middle  of  a  very 
narrow  slit  by  means  of  SQ.  This  done,  rotate  the  prism  by  turning  £5, 
and  see  if  the  slit  can  be  moved  by  it  at  all  in  the  direction  of  decreasing 
deviation.  If  so,  move  the  cross  hair  up  to  the  new  position  by  means  of  S6. 
Finally,  compute  the  index  of  refraction  from  equation  9. 


EXAMPLE 

I.  The  scale  of  the  graduated  circle  of  the  spectrometer  used  was  divided 
into  degrees  and  I  degrees  ;  that  is,  the  smallest  scale  division  was  20 
minutes.     The  vernier  was  divided  by  long  lines  into  20  divisions,  each  of 
which  was  divided  into  two  parts.     These  40  vernier  divisions  corresponded 
to  13  degrees,  that  is,  to  39  scale  divisions;  hence  one  vernier  division 
was  equal  to  f§  of  a  scale  division,  that  is,  each  vernier  division  lacked 
^-Q  of  being  a  whole  scale  division.     Hence  the  vernier  read  to  ^  of  20 
minutes,  or  to  30  seconds,  the  long  lines  on  the  vernier  indicating  minutes. 

The  readings  for  the  setting  on  the  face  P  were  315°  46'  30"  on  vernier  «, 
and  135°  46'  30"  on  vernier  b.  The  readings  for  the  setting  on  face  P/  were 
75°  44'  on  a,  and  255°  44'  on  vernier  b.  The  prism  table  was  "therefore 
rotated  through  119°  57'  30".  Hence  A  =  60°  2'  30". 

The  reading  on  vernier  a  at  M  (Fig.  196)  was  330°  36'  30",  and  that  at 
M'  210°  32'  30".  The  difference  is  120°  4'.  Hence  A  =  60°  2'.  The  mean 
yalue  of  A  was  therefore  60°  2'  15". 

II.  The  readings  for  the  angle  of  minimum  deviation  gave  D=  53°  6'. 

A  +  D      60°  2' 15"  + 53°  6' 
Hence  i  =  — ^—  =  -        —^-         -  =  56°  34'  8", 

A       60°  2' 15" 
and  r  =  —  = =  30°  1'  8". 

Therefore  n  =  —  —  =  1.6679. 

sin  r 


CHAPTEE  XXIV 
TOTAL  REFLECTION 

176.  Form  of  the  wave  front  in  the  second  medium.  In  con- 
nection with  the  discussion  of  the  formation  of  an  image  by  a  lens 
(see  sect.  155),  the  statement  was  made  that,  in  general,  a  wave 
which  is  spherical  in  one  medium  is  no  longer  spherical  after  it 
has  passed  into  a  second  medium.  The  correctness  of  this  assertion 


FIG.  199 

can  be  convincingly  shown  by  the  following  experiment.  If  a 
pencil  is  thrust  normally  into  a  body  of  water  and  viewed  by  an 
eye  which  looks  nearly  along  the  surface  (Fig.  199),  the  portion 
of  the  pencil  which  is  beneath  the  surface  will  appear  extremely 
short,  while,  if  the  eye  is  slowly  raised  from  position  1  to  position 
5,  the  length  of  the  submerged  portion  will  appear  to  become 
greater  and  greater.  This  means,  of  course,  that  the  center  of 

285 


286  ELECTRICITY,  SOUND,  AND  LIGHT 

curvature  of  the  wave  front  ABC  is  different  for  different  points 
on  ABC\  that  is,  the  curvature  is  not  constant.  In  other  words, 
the  wave  front  ABC  is  not  spherical. 

The  same  conclusion  is  reached  by  a  consideration  of  the  law 
of  refraction  deduced  in  the  last  chapter,  namely, 

velocity  in  air        sin  i 
n  =  -         — V-  —  • 

velocity  in  water      sm  r 

This  equation  shows  that  whenever  light  travels  from  a  medium 
of  lesser  speed  to  one  of  greater,  sin  i  is  greater  than  sin  r,  that  is, 
the  ray  emerging  into  the  medium  of  greater  speed  bends  away 
from  the  perpendicular.  Hence,  as  r  is  increased,  i  must  reach 
90°,  and  the  emerging  ray  lie  flat  along  the  surface  before  r 
reaches  90°.  If,  then,  this  last  ray  enters  an  eye  in  the  surface, 
the  point  beneath  the  surface  from  which  the  ray  comes  must, 
of  course,  appear  to  lie  in  the  surface  itself.  That  is,  as  the  eye 
is  raised  from  a  position  in  the  surface  through  positions  1,  2,  3, 
and  4  (Fig.  199),  the  point  from  which  the  light  comes  must 
appear  to  sink  from  the  surface  to  a  greater  and  greater  depth 
beneath  it,  precisely  as  it  was  observed  to  do  in  the  preceding 
experiment. 

177.  Caustics.    This  fact  of  the  modification  of  a  wave  front 
by  reflection  or  refraction  so  that  it  is  no  longer  spherical,  and 

therefore  has  not  a  single 
definite  focus,  or  center, 
after  such  modification, 
is  the  cause  of  a  group 
of  phenomena  known  as 
caustics.  These  can  be 
observed  most  easily  with 
-pIG  200  lenses  or  mirrors  of  large 

curvature.      One    of    the 

most  familiar  of  such  caustics  is  that  formed  by  a  cylindrical  lens 
such  as  a  tumbler  of  water.  When  the  sun's  rays  fall  upon  such 
a  vessel,  the  rays  which  pass  through  the  outer  portion  are  brought 
together  at  points  much  nearer  the  tumbler  than  are  those  which 
pass  through  the  central  portion.  In  other  words,  the  changes  in 


TOTAL  REFLECTION  287 

the  form  of  the  wave  front  which  take  place,  both  when  the  wave 
enters  the  lens  and  when  it  leaves  it,  are  such  that  the  outer  edges 
of  the  wave  which  has  passed  through  the  tumbler  have  their 
centers  of  curvature  relatively  close  to  the  tumbler,  while  the 
center  of  curvature  of  the 
central  portion  of  the 
wave  is  more  remote  (see 
Fig.  200).  The  caustics 
are  the  curved  lines  ab, 
cb,  which  represent  the 
envelopes  of  the  radii  of 
curvature  of  the  different 
portions  of  the  wave  (see 
also  ab,  Fig.  199).  Fig- 
ure 201  shows  a  caustic  FlG  2oi— 
formed  by  reflection  from 

a  spherical  mirror  of  very  large  aperture.  It  will  be  seen  from  the 
figure  that  a  spherical  mirror  cannot  be  said  to  have  a  definite 
focus  unless  its  aperture  is  small,  that  is,  unless  the  angle  sub- 
tended at  F  by  the  mirror  does  not  exceed  fifteen  or  twenty  degrees. 
The  only  general  case  of  the  modification  of  a  spherical  wave  by 
reflection  or  refraction,  in  which  the  modified  wave  has  a  strictly 

spherical  form,  is  the  case 
of  reflection  from  a  plane 
surface  (Fig.  169,  p.  246). 
Nevertheless,  for  a  given 
Ly  form  of  incident  wave, 

for  example  a  plane  wave, 


.£^  it  is  always   possible  to 

FIG  202  give  such  a  shape  to  the 

reflecting    or    refracting 

surface  as  to  make  the  resulting  wave  front  spherical.  Thus  when 
a  plane  wave  falls  upon  a  parabolic  mirror  the  reflected  wave  front 
is  strictly  spherical,  and  hence  no  caustic  is  formed  (see  Fig.  202). 
Conversely,  when  a  wave  originates  at  the  focus  of  a  parabolic 
mirror  it  is  reflected  as  a  rigorously  plane  wave,  that  is,  as  a  parallel 
beam.  Hence  the  use  of  parabolic  mirrors  in  search  lights. 


288 


ELECTRICITY,  SOUND,  AKD  LIGHT 


The  lack  of  sphericity  in  the  waves  which  have  passed  through 
ordinary  lenses  bounded  by  spherical  surfaces  is  seen  in  the  fact  that 
it  is  impossible  with  such  lenses  to  obtain  images  in  which  there  is 
sharp  definition ;  for  the  rays  which  pass  through  the  edges  of  the 
lens  are  brought  to  a  focus  nearer  to  the  lens  than  are  the  rays 
which  pass  through  the  middle  portion.  This  phenomenon  of  caustics, 
as  shown  by  ordinary  lenses,  is  usually  known  as  spherical  aberration. 
It  may  be  reduced  by  decreasing  the  aperture  of  the  lens,  but  this 
also  decreases  its  resolving  power  (see  note  *,  p.  251).  It  is  elimi- 
nated entirely  by  special  combinations  of  convex  and  concave  sur- 
faces, or  by  grinding  the  surfaces  of  a  single  lens  into  special  form. 
178.  Total  reflection.  From  the  law  of  refraction  stated  in  sec- 
tion 176,  it  will  be  seen  that  when  a  ray  of  light  travels  from  a  me- 
dium of  lesser  speed  to  one  of  greater,  if  we  continue  to  increase 

r  after  i  has  become  90°, 
there  can  be  no  refracted 
ray  at  'all.  Experiment 
completely  confirms  this 
conclusion  and  shows  that 
light  which  is  incident 
upon  the  upper  medium 
(Fig.  203)  at  an  angle 
greater  than  that  for  which 
t  =  90°is  totally  reflected 
in  accordance  with  the 
usual  law  of  reflection, 
namely,  the  angle  of  incidence  r  equals  the  angle  of  reflection  r'. 

Figure  203  shows  the  course  of  various  rays  starting  at  different 
angles  from  the  point  /  beneath  a  surface  of  water.  The  value  of  r 
for  which  i  =  90°,  and  hence  the  value  of  r  beyond  which  total  reflec- 
tion takes  place,is  called  the  critical  angle  (see  c,Fig.  203).  It  depends 
upon  the  relative  index  of  refraction  of  the  two  media.  Thus  since 

sin  i 


FIG.  203 


n  = 


smr 


it  is  evident  that,  since  sin  90°  =  1,  for  c,  the  critical  angle,  there 
is  the  following  relation :  j 


sine 


TOTAL  REFLECTION  289 

Thus  for  water,  for  which  n  =  1.33,  sin  c  =  1/1.33,  or  c  =  48.5°. 
This  shows  that  no  ray  of  light  which  comes  from  water  to  a 
surface  separating  water  and  air  so  as  to  make  with  the  normal 
an  angle  greater  than  48.5°  can  pass  out  into  the  air. 

179.  Total-reflection  prism.    An  application  of  the  principle  of 
total  reflection  is  made  in  the  construction  and  use  of  the  so-called 
total-reflection  prism,  a  device  for  changing  the  direction  of  a  beam 
of  light  by  90°  without  sensibly  diminishing  its  intensity  or  pro- 
ducing in  it  dispersion,  if  it  is  a  beam 

of  white  light.    The  index  of  refraction 

of  practically  all  forms  of  glass  is  more 

than   1.5.    Now   for   a    substance    for 

which  n  is  1.5,  c  is  42°.    Hence  if  a 

beam  of  light   op  (Fig.  204)  enters   a 

right-angled  prism  at  normal  incidence, 

it  will  strike  the  face  AB  at  45°,  that 

is,  at  an  angle  greater  than  the  critical 

angle.    It  will  therefore  be  totally  reflected  and  pass  out  normally 

through  the  face  AC  without  refraction  or  dispersion. 

180.  Determination  of  the  index  of  refraction  from  measure- 
ments upon  the  critical  angle.    One   of   the   most   satisfactory 
methods  of  demonstrating  the  facts  of  total  reflection,  and  at  the 
same  time  of  determining  the  index  of  refraction  of  a  substance 
which  can  be  put  into  the  form  of  a  prism  with  three  polished 
faces,  is  as  follows.    Let  MON  (Fig.  205)  be  any  broad  source  of 
monochromatic  light,  CDE  a  prism  so  set  that  light  from  MOX  is 
refracted  and  reflected  to  the  telescope  T  in  the  manner  shown  in 
the  figure.    The  different  rays  which  come  to   T  from  different 
points  on  MON  will  obviously  strike  the  surface  CD  at  different 
angles.    Those  rays  which  strike  CD  at  an  angle  greater  than  the 
critical  angle  will  be  totally  reflected,  while  those  which  strike  at 
an  angle  less  than  the  critical  angle  will  be  partially  reflected  and 
partially  transmitted.    If  OpqS  is  the  ray  which  strikes  exactly  at 
the  critical  angle,  then  all  of  the  light  which  comes  to  T  from  the 
portion  Cq  of  the  face  CD  will  have  struck  CD  at  angles  greater 
than  the  critical  angle,  and  hence  it  will  have  undergone  total 
reflection.    But  all  of  the  light  which  comes  to  T  after  reflection 


290 


ELECTRICITY,  SOUND,  AND  LIGHT 


at  qD  will  have  struck  CD  at  angles  less  than  the  critical  angle, 
and  hence  it  will  have  undergone  only  partial  reflection,  the  other 
part  having  been  transmitted.  Hence  the  surface  CD  should 
appear  to  consist  of  two  parts  of  unequal  illumination,  the  portion 
Cgr,  which  corresponds  to  total  reflection,  appearing  brighter  than 

the  portion  qD,  which 
corresponds  to  partial 
reflection.  The  line 
of  junction  of  these 
two  portions  is  quite 
sharply  marked,  so 
that  the  precise  point 
q  at  which  total  re- 
flection begins  can  be 
accurately  determined. 
Suppose  that  the  cross 
hairs  of  the  telescope  T 
are  first  set  upon. this 
point  and  the  telescope 
then  rotated  until  it  is 
at  right  angles  to  the 
face  ED,  that  is,  sup- 
pose that  the  angle  i  is 

accurately  measured.  The  index  of  refraction  may  then  be  obtained 
from  i  and  the  angle  A  of  the  prism  as  follows.  We  have 


FIG.  205 


sin  i 
sinr 

I 

sin  c 

A  =  r  +  c. 


(2) 

(3) 


From  these  equations  we  have  only  to  eliminate  c  and  r  in  order 
to  obtain  n  in  terms  of  i  and  A.    From  (1)  and  (3)  we  obtain 


sin  i 


sin 


sin  (A  —  c)      sin  A  cos  c  —  cos  A  sin  c 


(4) 


TOTAL  REFLECTION  291 

From  equation  (2)  sin  c  =  1/n,  and  also 
cosc=vl —  sin2c  = 

Substitution  of  these  values  in  equation  (4)  gives 

sin  i 

(5) 


sin  A  •  -  V?t'2  —  1  —  cos  A  •  — 
n  n 

2  /sin  i  +  cos  A\* 

or  7i2-l  =  ( —       -).  (6) 

\       sin  A       / 

^       £  I  /sin  i  +  cos  A\2 

Therefore  n  =  A  ( -  -1+1.  (7) 

N  \       sin  A       I 

EXPERIMENT  24 

Object.  To  find  by  a  total-reflection  method  the  index  of  refraction  of 
the  prism  used  in  Experiment  23. 

Directions.  Focus  the  telescope  for  parallel  rays  and  set  its  axis  at  right 
angles  to  the  axis  of  rotation  by  the  method  of  Experiment  23.  Then  set 
up  a  broad  sodium  flame  AION  (Fig.  205)  on  the  level  of  the  telescope  of 
the  spectrometer  and  at  a  distance  from  it  of  two  or  three  feet.  Turn  the 
telescope  and  prism  into  about  the  relative  positions  shown  in  the  figure, 
or  until  the  image  of  the  flame  reflected  from  the  face  CD  can  be  seen  in 
the  telescope.  Then  rotate  prism  and  telescope  together  until  the  yellow 
line  which  divides  the  field  into  two  parts  of  unequal  intensity  is  seen.  Set 
the  cross  hairs  upon  this  line,  and  read  the  vernier  and  scale.  Without 
moving  the  prism  rotate  the  telescope  into  a  position  at  right  angles  to  the 
face  DE,  as  determined  by  the  coincidence  of  the  two  images  of  the  cross 
hairs  as  seen  in  the  Gauss  eyepiece ;  read  again  the  vernier  and  thus  deter- 
mine the  angle  i.  From  this  value  of  i  and  from  the  value  of  A  as  obtained 
in  Experiment  23,  find  n  by  equation  (7).  Compare  the  value  of  n  thus  found 
with  that  obtained  in  Experiment  23. 

EXAMPLE 

The  reading  of  the  vernier  for  the  setting  on  the  line  of  total  reflection 
was  303°  54'.  The  reading  for  normal  incidence  was  262°  46'.  Hence 
i  =  41°  8'.  The  angle  A  as  found  in  Experiment  23  was  60°  2'  15".  Hence 


+  1  =  1.6683. 


sin  60°  2'  15" 

The  value  found  in  Experiment  23  was  1.6679.    The  difference  is  .02 
per  cent. 


CHAPTEE   XXV 
PHOTOMETRY 

181.  Laws  of  illumination.  When  a  surface  is  illuminated  by 
light  from  a  luminous  point  we  may  define  the  intensity  of  illu- 
mination as  the  quantity  of  luminous  energy  which  falls  upon 
the  surface  per  second  divided  by  the  area  of  the  surface,  that  is, 
as  the  quantity  of  light  per  unit  area.  It  will  be  obvious  at  once 
from  this  definition  that  if  we  consider  two  surfaces  at  a  given 
distance  from  the  point,  the  one  normal  to  the  direction  of  propa- 
gation of  the  light  and  the  other  so  inclined  that  its  normal 


EIG.  206 

makes  an  angle  6  with  this  direction,  then  the  illumination  72 
of  the  inclined  surface  is  related  to  the  illumination  /x  of  the 
normal  surface  by  the  equation 


(1) 

For,  since  the  same  energy  E  falls  upon  the  inclined  surface  s2 
(Fig.  206)  as  upon  the  normal  surface  slt  we  have  Il  =  E/sl  and 
I  =  Es  hence 


Again,  the  illumination  upon  two  surfaces,  s1  and  s2,  at  dis- 
tances P!  and  r2  respectively  from  a  given  point  source,  and 
making  the  same  angle  with  the  direction  of  propagation  of  the 
disturbance,  are  related  by  the  equation 

i  __  r2  .  /o\ 

/  ~v; 

^2  '  \ 

292 


PHOTOMETEY  293 

for,  since  the  same  energy  E  falls  upon  the  surface  s2  (Fig.  207)  as 
upon  the  surface  slt  we  have  1^  =  E/sv  and  /2  =  E/s2.    Hence 


The  two  results  thus  obtained  may  be  stated  as  follows :  Intensity 
of  illumination  is  inversely  proportional  to  the  square  of  the  dis- 
tance from  the  point  source,  and  directly  proportional  to  the  cosine 
of  the  angle  which  the  normal  to  the  illuminated  surface  makes 
with  the  direction  of  the  incident  light. 

This  conclusion  has  followed  simply  from  the  definition  which 
we  have  given  to  the  word  illumination.  If  this  definition  is  to 
have  any  practical  value  in  photometry,  it  is  necessary  that  two 
surfaces  appear  to  the  eye  equally  bright  whenever  they  are  illu- 
minated with  equal  intensities,  as  here  defined.  Experiment  shows 


FIG.  207 


that  this  is  the  case  when,  and  only  when,  the  two  surfaces  are 
illuminated  by  lights  of  the  same,  or  nearly  the  same,  color.  For 
example,  a  screen  placed  at  the  distance  of  one  meter  from  a 
single  candle  appears  exactly  as  bright  as  a  similar  screen  placed 
at  a  distance  of  two  meters  from  four  precisely  similar  candles 
placed  very  close  together. 

182.  Intensities  of  different  sources  of  light.  The  example  just 
given  suggests  at  once  a  method  of  comparing  the  quantities  of 
light  emitted  by  different  sources  of  the  same  color.  For  we  have 
only  to  arrange  the  sources  so  that  each  illuminates  at  the  same 
angle  one  of  two  adjacent  surfaces,  and  then  to  vary  the  distances 
of  the  two  lights  from  their  respective  screens  until  these  two 
screens  appear  to  have  equal  illumination.  It  will  then  be  seen 
at  once  from  the  example  of  the  preceding  section  that  the  inten- 
sities of  the  two  sources,  that  is,  the  quantities  of  light  emitted  by 
them,  are  directly  proportional  to  the  squares  of  their  respective 
distances  from  the  equally  illuminated  surfaces.  Algebraically 


294  ELECTRICITY,  SOUND,  AND  LIGHT 

stated,  if  Ll  and  L2  represent  the  intensities  of  the  sources,  and 
dl  and  d2  their  respective  distances  from  equally  illuminated  screens, 
then  we  have  , 


183.  Photometric  standards.    The  unit  of  light-emitting  power 
first  used  for  a  comparison  of  the  intensities  of  different  sources 
of  light  was  the  candle.    Thus,  a  light  of  sixteen  candle  power  is 
one  which  produces  the  same  intensity  of  illumination  at  a  dis- 
tance of  four  meters  as  does  a  single  candle  at  a  distance  of  one 
meter.     Since  candles  of  different  composition  and  size,  burning 
under  different  conditions,  differ  widely  in  the  amounts  of  light 
which  they  emit,  it  is  obviously  necessary  to  specify  the  type  of 
candle  to  be  used  as  a  standard.    The  so-called  normal  candle  is  a 
candle  of  paraffin,  2  cm.  in  diameter  and  burning  with  a  flame 
50  mm.  high.    In  Great  Britain  the  legal  standard  of  light  is  a 
sperm  candle  which  burns  7.776  g.  of  spermaceti  per  hour.     In 
Germany  the  standard  candle  has  been  replaced  by  a  special  form 
of  lamp  invented  by  Hefner-  Alteneck.    It  burns  amyl  acetate,  and, 
when  regulated  so  as  to  have  a  flame  40  mm.  high,  emits  one 
so-called  Hefner  unit  of  light.    This  is  equivalent  to  .81   of  a 
normal    candle.    Another    special    form   of   oil    lamp,   called    the 
Carcel  standard,  is  in  use  in  France.    It  is  equivalent  to  about  9.4 
normal  candles.    For  many  purposes  it  is  very  convenient  to  use 
a  carbon-filament  glow  lamp  as  a  standard,  since  with  suitable 
precautions  it  emits  a  very  constant  light. 

184.  The   Lummer-Brodhuii  photometer.    The  most  approved 
modern    instrument    for    comparing   the  intensities    of    different 
sources  of  light  is  the  Lummer-Brodhun  photometer.    The  sur- 
faces the  illumination  of  which  by  the  two  sources  Sl  and  Sz 
(Fig.  208)  is  made  the  same,  are  the  opposite  sides  of  a  white 
opaque  screen  AB.    These   surfaces  are  viewed  by  an  eye  at  E 
with  the  aid  of  two  plain  mirrors  M^  and  Mz.    In  order  to  bring 
the  two  sides  of  AB  into  immediate  juxtaposition,  as  seen  by  the 
eye  at  E,  the  principle  of  total  reflection  is  made  use  of  in  the 
construction  of  the  prism  CD.    This  consists  of  two  right-angled 
prisms,  CGH  and  DGH,  pressed  very  firmly  together  along  the 


PHOTOMETRY 


295 


faces  GH,  which  are  made  so  as  to  come  into  perfect  contact  in 
certain  places,  "but  not  to  come  into  contact  in  other  places.  Now 
the  light  which  conies  to  E  through  the  portion  of  the  interface 
GH  in  which  the  surfaces  are  in  perfect  contact  is  light  which 
comes  from  the  left  side  of  AB,  undergoes  reflection  at  Mv  and 
then  passes  without  change  of  medium  through  the  prism  from 
the  face  CG  to  the  face  HD.  On  the  other  hand,  the  light  which 
comes  to  E  from  the  portions  of  the  face  GH  which  are  not  in 
perfect  contact,  that  is,  from  places  at  which  an  air  film  exists 
between  the  two  surfaces,  is  composed  entirely  of  rays  which  have 
come  from  the  right 
side  of  AB  by  way 
of  the  mirror  J/2,  and 
have  then  undergone 
total  reflection  at  the 
surface  of  the  air 
film.  Hence,  if  the 
two  sides  of  AB  are 
exactly  similar  sur- 
faces, and  if  M^  and 
M0  are  exactly  simi- 
lar mirrors,  it  is  only 
necessary  to  set  AB 
at  such  a  point  be- 
tween the  sources  Sl 
and  $,  that  the  whole 
surface  GH,  as  seen  from  E,  is  of  uniform  illumination,  and  then  to 
apply  equation  (3).  In  order  to  eliminate  any  possible  inequalities 
in  the  two  sides  of  AB,  or  in  the  mirrors  M^  and  J/2,  the  whole 
instrument  is  usually  rotated  through  180°  about  an  axis  passing 
through  AB.  This  interchanges  the  two  sides  of  AB  and  also  the 
mirrors  J/x  and  J/2.  The  mean  of  the  settings  before  and  after 
reversal  is  then  taken  as  the  correct  setting.  Figure  209  shows 
a  horizontal  section  of  the  instrument. 

185.  Photometric  values  of  lights  of  different  colors.  As  stated 
in  section  181,  such  an  instrument  as  that  described  in  the  last 
section  is  capable  of  yielding  concordant  results  only  when  the 


296 


ELECTBICITY,  SOUND,  AND  LIGHT 


two  sources  are  of  the  same  color.  If  the  colors  of  the  sources 
differ,  the  impression  of  color  contrast  is  so  strong  that  the  setting 
for  equality  of  illumination  becomes  an  estimate  which  even  the 
same  person  cannot  duplicate  closely,  and  upon  which  different 
persons  will  have  widely  different  judgments.  If  we  are  to  con- 
tinue to  define  illumination,  as  in  section  181,  as  the  quantity  of 
luminous  energy  per  unit  area,  it  would,  of  course,  be  possible  to 
compare  the  intensities  of  ejnission  of  lights  of  different  color, 
such  as  red  and  blue,  by  allowing  the  radiations  from  the  different 
sources  to  fall  at  a  given  distance  upon  equal  surfaces  which  com- 
pletely absorb  them  both,  and  then  measuring  the  relative  amounts 

of  heat  developed  in  these  sur- 
faces per  second.  This  would 
presuppose,  of  course,  the  elimi- 
nation of  the  heating  effects 
due  to  the  nonluminous  radi- 
ations, which,  in  general,  ac- 
company luminous  radiations. 
This  might  be  accomplished  by 
passing  the  lights  from  both 
sources  through  a  prism,  and 
allowing  only  the  visible  por- 
tions of  the  spectra  to  fall  upon 

the  comparison  screens  whose  change  in  temperature  was  to  be 
observed.  Since,  however,  even  within  the  limits  of  the  visible 
spectrum,  surfaces  illuminated  with  equal  energy  do  not,  in  general, 
appear  equally  bright,  it  is  obvious  that  such  a  comparison  would 
give  us  no  information  as  to  the  relative  values  for  the  purposes 
of  vision  of  different  lights. 

Photometry,  then,  as  an  accurate  science,  is  limited  by  the  very 
nature  of  the  eye  to  the  comparison  of  lights  of  approximately  the 
same  color.  In  the  case  of  complex  lights,  such  as  most  commer- 
cial lamps  produce,  we  must,  in  general,  be  content  with  rough 
approximations  in  our  estimations  of  relative  candle  power,  for 
such  lamps  generally  differ  considerably  in  color.  Nevertheless,  as 
suggested  above,  it  is  always  possible  to  pass  both  of  the  lights  to 
be  compared  through  a  prism  and  thus  to  separate  each  into  its 


209 


PHOTOMETRY  297 

constituent  colors,  after  which  the  relative  intensities  of  any  par- 
ticular color  in  the  two  sources  may  be  accurately  compared.  This 
process  is  called  spectro-photometry. 

186.  The  laws  of  radiation.  The  fact  that  even  when  a  body 
is  placed  in  the  best  obtainable  vacuum  its  temperature  continually 
falls  when  it  is  surrounded  by  a  colder  body,  such,  for  example,  as 
liquid  air,  shows  that  all  bodies  are  at  all  temperatures  continually 
radiating  energy  in  the  form  of  ether  waves.  It  follows  that  when 
a  body  is  at  constant  temperature  it  must  be  absorbing  energy  at 
precisely  the  same  rate  at  which  it  is  radiating  energy.  This  prin- 
ciple is  known  as  Prevost's  law  of  exchanges. 

How  the  total  amount  of  radiated  energy  varies  with  the  tem- 
perature, and  how  it  is  distributed  among  the  waves  of  different 
wave  length  are  questions  which  have  been  made  the  subjects  of 
many  important  investigations,  both  experimental  and  theoretical. 
In  1879  Stefan,  of  Vienna,  discovered  experimentally  that  when  a 
black  body,  such  as  a  carbon  filament,  is  heated  to  different  tem- 
peratures the  total  intensity  of  emission  is  directly  proportioned  to 
the  fourth  power  of  the  absolute  temperature.  This  is  known  as 
Stefan's  law  of  radiation.  It  has  since  been  deduced  from  purely 
theoretical  considerations.  It  holds  strictly  only  for  black  bodies, 
that  is,  bodies  which  absorb  all  radiations  which  fall  upon  them, 
but  is  approximately  correct  for  most  solids. 

"With  reference  to  the  distribution  of  the  emitted  energy  among 
the  different  wave  lengths  some  information  may  be  obtained  from 
very  familiar  experiments.  It  is  a  matter  of  common  observation 
that  as  the  temperature  of  any  solid  is  continuously  raised  it  at 
first  emits  only  heat  waves,  that  then  visible  waves  of  a  very  dull 
red  color  make  their  appearance,  and  that  the  color  then  changes 
first  to  orange,  then  to  yellow,  and  finally  to  a  brilliant  white. 
This  behavior  shows  that  as  the  temperature  is  raised  shorter  and 
shorter  wave  lengths  are  added  to  the  emitted  light.  It  must  not 
be  supposed,  however,  that  a  white-hot  body  emits  less  red  light 
than  does  a  red-hot  one.  In  general  the  intensity  of  emission  of 
all  wave  lengths  increases  rapidly  with  temperature,  but  the 
rate  of  increase  is  more  rapid  in  the  case  of  the  shorter  waves. 
If  the  light  is  passed  through  a  prism  and  the  energy  of  radiation 


298  ELECTRICITY,  SOUND,  AND  LIGHT 

measured  in  the  different  portions  of  the  spectrum  hy  means  of 
its  heating  effect  upon  a  suitable  thermometer,  the  wave  length  at 
which  the  heating  is  a  maximum  continually  shifts  toward  shorter 
and  shorter  wave  lengths  as  the  temperature  rises.  A  law  known 
as  the,  displacement  law  has  been  brought  to  light  by  the  most 
careful  experimental  and  theoretical  investigation.  It  asserts  that 
the  wave  length  \m,  at  which  the  heating  is  a  maximum,  so  shifts 
with  rising  temperature  that  the  product  of  \m  by  the  absolute  tem- 
perature T  is  a  constant.  Thus  at  820°  absolute  the  wave  length 
at  which  the  radiated  energy  is  a  maximum  is  .00356  mm.,  while 
at  1640°  absolute  \m  is  .00178.  In  symbols  the  law  is 

\mT  =  constant. 

The  enormous  increase,  however,  in  the  intensity  of  radiation  of 
any  particular  wave  length  with  temperature  is  shown  by  the  fol- 
lowing example.  If  the  intensity  of  the  red  light  ( A,  = . 000656  mm.) 
emitted  by  a  body  at  1000°  C.  is  called  1,  at  1500° C.  the  intensity 
of  the  same  wave  length  is  over  130,  and  that  at  2000°  C.  over 
2100. 

All  bodies  in  which  the  radiation  of  heat  and  light  is  unaccom- 
panied by  permanent  chemical  or  molecular  changes  of  any  sort 
are  found  to  become  visible  at  the  same  temperature,  namely  at 
about  525° C.,  and  to  become  white  hot  at  about  1200° C.  Never- 
theless the  total  intensity  of  emission  depends  not  only  upon  the 
temperature,  but  also  upon  the  nature  of  the  radiating  body.  At  a 
given  temperature,  however,  no  body  emits  any  wave  length  in 
greater  intensity  than  does  a  black  body.  These  statements  apply 
only  to  radiations  produced  by  temperature  alone.  When  the  radi- 
ation is  produced  by  molecular  or  chemical  changes  it  is  of  a 
different  type,  called  luminescence.  It  is  illustrated  by  the  electri- 
cal discharge  in  vacuum  tubes,  by  the  glowworm  light,  and  other 
similar  phenomena.  In  cases  of  luminescence  there  is  often  a  strong 
emission  of  light  with  very  little  heat. 

187.  Optical  efficiency.  It  will  be  seen  from  the  preceding 
paragraph  that  if  we  are  to  use  temperature  radiation  for  the 
purposes  of  commercial  lighting,  then  the  chief  requisite  of  the 
incandescent  body. is  that  it  be  capable  of  withstanding  a  very 


PHOTOMETRY  299 

high  temperature.  Thus  the  chief  advantage  of  the  arc  light  over 
the  incandescent  electric  light  is  that  in  the  former  the  carbon  is 
in  such  form  that  it  can  be  raised  to  the  extreme  temperature  of 
3800°  C.,  while  the  slender  filament  of  the  latter  permits  a  tempera- 
ture of  only  about  1900°C.  Similarly,  the  high  efficiencies  of  the 
"VVelsbach  and  the  Nernst  lights  are  due  to  the  fact  that  in  both 
of  them  the  incandescent  body  is  raised  to  a  temperature  of  about 
2300°C.* 

The  optical  efficiency  of  a  source  of  light  is  defined  as  the  ratio  of 
the  luminous  energy  radiated  per  second  to  the  energy  required  to 
maintain  the  light  for  this  time.  In  the  ordinary  oil  or  gas  light 
not  more  than  1  per  cent  of  the  total  heat  energy  produced  by  the 
combustion  is  represented  in  the  luminous  radiations.  The  electric 
light  is  much  more  efficient.  An  ordinary  incandescent  lamp  con- 
sumes about  3.5  watts  per  candle  power.  The  luminous  energy 
radiated  per  second  by  a  light  of  one  candle  power  is  about 
1.3  X  106  ergs.  Since  a  watt  is  107  ergs  per  second,  the  efficiency  e 
of  the  incandescent  lamp  is  given  by 

1.3xl06 


In  the  best  arc  lamps  the  efficiency  is  as  high  as  one-  third  watt  per 
candle  power,  or  approximately  ten  times  that  of  the  incandescent 
light.  This  means  that  in  such  an  arc  light,  if  we  neglect  the  con- 
sumption of  the  carbons,f  as  much  as  37  per  cent  of  the  total  energy 
expended  is  utilized  in  the  production  of  light.  These  figures  relate 
to  the  new  flaming  arc  produced  between  carbons  which  are  im- 
pregnated with  some  salt  like  calcium  fluoride,  and  in  which  the 
light  comes  largely  from  the  incandescent  vapors  in  the  arc  itself. 
The  efficiency  of  the  ordinary  arc,  in  which  the  light  comes  chiefly 
from  the  luminous  center  of  the  positive  carbon,  is  in  general  not 
greater  than  1.2  watts  per  mean  spherical  candle  power. 

The  efficiency  of  the  Nernst  lamp  is  about  twice  that  of  the 
incandescent  ;  that  of  the  Cooper-Hewitt  light  about  six  times  that 

*  It  should  be  said,  however,  that  the  arc  light  is  probably  not  an  example 

of  pure  temperature  radiation.    The  light  is  due  in  small  part  to  luminescence. 

t  The  neglecting  of  this  factor  obviously  renders  the  result  quite  uncertain. 


300  ELECTRICITY,  SOUND,  AND  LIGHT 

of  the  incandescent.  The  new  tungsten  and  tantalum  lamps  have 
about  twice  the  efficiency  of  the  carbon  filament,  the  gain  being 
wholly  due  to  the  higher  temperatures  employed. 

In  rating  the  efficiency  of  an  electric  light  it  is  not  customary 
to  reduce  to  absolute  units,  as  was  done  above,  but  merely  to  give 
the  ratio  of  the  candle  power  produced  to  the  watts  consumed. 

EXPERIMENT  25 

Object.    To  plot  the  efficiency  curve  of  an  incandescent  lamp. 

Directions.  Arrange  a  Lummer-Brodhun  photometer  P,  a  16-candle- 
power,  50-volt  lamp  /,  an  ammeter  A,  reading  to  5  amperes,  a  voltmeter  F, 
a  variable  resistance  R,  and  a  battery  or  dynamo  B,  as  in  Figure  210.  Then 
connect  a  16-candle-power,  110-volt  lamp  L  to  the  lamp  socket,  and,  using 
L  as  a  standard,  find  the  candle  power  of  I  for  a  series  of  values  of  the 
P.I),  across  its  terminals.  Beginning  with  a  P.D.  of  40  volts,  decrease  R 
and  thus  increase  P.D.  by  about  3  volt  steps  until  65  is  reached,  taking- 
readings  on  the  ammeter,  voltmeter,  and  photometer  at  every  change. 


/ 

L 

50  If                                                   f> 

no  v. 

<$> 

V 

n                  1 

IS££5T>« 

770  V. 

Compute  in  every  case  the  number  of  watts  (volts  x  amperes)  required  to 
produce  one  candle  power  of  illumination,  and  then  plot  a  curve  in  which 
abscissas  represent  volts  and  ordinates  represent  optical  efficiencies  (candle 
power  divided  by  watts).  Tabulate  the  data  in  one  corner  of  the  sheet  and 
let  the  graph  constitute  the  record  of  the  experiment. 


EXAMPLE 


The  record  of  this  experiment  is  shown  in  Figure  211.  It  is  seen  that 
the  efficiency  continuously  increases  with  the  P.D.  and  would  doubtless 
continue  to  do  so  until  the  lamp  burned  out.  The  reason  that  the  lamp  is 
not  run  at  a  higher  P.D.  than  that  marked  upon  it  is  that  the  increased 
optical  efficiency  is  more  than  offset  by  the  decreased  life  of  the  lamp. 


PHOTOMETRY 


301 


Ds.  C.P.  Eff. 

122.  6.54  .145 

115.5  8.60  .180 

106.5  12.33 

19.54  .309 

91.0  23.42  .345 

82.0  33.12  .443 

73.5  47.39  .579 

58.5  93.61 


54.6  93.5 
63.3  105.0 
67.8  109.0 


1.25  74.7  118.0 
81.9  126.5 
1.45  101 .5  141.5 


FIG.  211 


CHAPTEE  XXVI 
DISPERSION  AND   SPECTRA 

188.  Newton's  experiments  on  dispersion.  It  was  in  the  year 
1669  that  Newton,  at  the  age  of  twenty-five,  published  his  justly 
celebrated  experiments  on  the  analysis  and  synthesis  of  white 
light,  —  experiments  which  during  more  than  two  centuries  formed 
the  basis  of  all  explanations  of  the  phenomena  of  color. 

These  experiments  consisted  in  admitting  light  through  a  small 
aperture  A  (Fig.  212)  into  a  darkened  room  and  observing  that  the 
round  image  BC  of  the  sun  which  was  produced  on  the 
wall,  before  the  prism  P  was  placed  in  the  path  of  the  beam, 
became  replaced  upon  the  interposition  of  the  prism 
by  the  band  of  colors  RV.  This  band  was  red  at  |§ 
the  end  which  corresponded  to  the  smallest  amount 


7 

A 


FIG.  212 


of  refraction,  and  changed  through  yellow,  green,  and  blue, 
into  violet  at  the  other  end.  Newton  further  placed  a  second 
prism  in  the  path  of  the  colored  band  in  the  position  indicated  by  P' 
of  the  figure,  and  observed  that  the  colors  wrere  perfectly  recornbined 
on  the  wall  into  white  light.  He  also  showed  that  it  was  impos- 
sible by  means  of  a  second  prism  to  further  decompose  any  one  of 
the  spectral  colors  into  more  elementary  parts. 

302 


DISPERSION  AND  SPECTRA  303 

In  view  of  these  experiments  it  has  been  customary,  since 
the  time  of  Newton,  to  regard  white  light  as  composed  of  a 
mixture  of  elementary  colored  lights  of  every  conceivable  wave 
length  between  that  of  the  longest  red  and  that  of  the  shortest 
violet.  As  a  matter  of  fact,  Newton's  experiments  show,  not  that 
white  light  actually  consists  of  all  of  these  colored  lights,  but 
merely  that  white  light  is  decomposed  by  a  prism  into  these 
colored  lights,  and  that  by  recombining  these  colors  we  do  actu- 
ally reproduce  upon  the  retina  the  effect  of  white  light.  How- 
ever, we  are  led  into  no  conclusions  which  are  at  variance  with 
experiment  if  we  adopt  Newton's  view  point  as  to  the  nature  of 
white  light,  and  we  shall  therefore  make  this  viewpoint  the  basis 
of  much  of  our  reasoning.  We  shall,  however,  return  to  a  more 
critical  analysis  of  this  subject  in  a  later  section  (see  sect.  195). 

Since  violet  light  is  refracted  more  than  red  light,  and  since 
the  amount  of  refraction  is  a  measure  of  the  change  of  velocity  in 
going  into  a  new  medium,  it  is  clear  that  the  shorter  visible  waves 
undergo  a  greater  change  of  velocity  in  going  into  glass  than  do 
the  longer  waves.  In  other  words,  the  velocity  of  propagation  of 
violet  light  through  glass  is  less  than  that  of  red  light.  The  exact 
ratio  of  these  velocities  is  the  ratio  of  the  indices  of  refraction  of 
glass  for  red  and  violet.  This  for  flint  glass  is  about  1.62/1.67. 

189.  Pure  spectra.  Since  Newton's  spectrum  consisted  merely 
of  a  row  of  circular  images  of  the  sun  in  different  colors,  and 
since  these  images  overlapped,  as  is  seen  from  a  consideration 
of  Figure  212,  it  is  evident  that  the  color  at  any  given  point 
of  this  spectrum  wTas  a  combination  of  two  or  more  colors ;  that 
is,  this  spectrum  did  not  consist  of  colors  each  of  which  corre- 
sponded to  one  particular  wave  length.  It  was  on  account  of  this 
fact  that  Newton  failed  to  notice  some  of  the  most  interesting 
characteristics  of  the  solar  spectrum ;  such,  for  example,  as  the 
Fraunhofer  lines  (see  sect.  192).  In  order  to  obtain  a  pure  spectrum 
it  is  necessary  to  avoid  this  overlapping  of  the  images  of  the  aper- 
ture in  different  colors.  But  since  a  small  aperture  of  any  shape 
whatever  will  always  produce  a  round  image  of  the  sun  at  BC,  in 
order  to  obtain  a  pure  spectrum  two  alterations  must  be  made 
in  Newton's  arrangement.  First,  the  aperture  must  be  a  very 


304  ELECTEICITY,  SOUND,  AND  LIGHT 

narrow  slit ;  and  second,  a  lens  must  be  placed  at  such  a  point 
between  the  aperture  and  the  screen  as  to  form  upon  the  screen, 
not  an  image  of  the  sun,  but  an  image  of  this  narrow  slit.  In  no 
other  way  can  the  image  in  any  particular  color  be  made  a  mere 
line.  When,  however,  the  slit  and  the  screen  are  at  conjugate  foci 
of  a  lens,  as  in  Figure  213,  the  spectrum  becomes  simply  a  row 
of  adjacent  line  images  in  different  colors  of  the  line  source. 
A  spectrum  formed  in  this  way  is  called  a  pure  spectrum. 

When  a    spectroscope  is  in  such  adjustment  that  an  image  of 
the  slit  is  formed  in  the  focal  plane  of  the  eyepiece  of  the  tele- 
scope, it  is  evident  that  a  pure  spectrum  may  be  obtained. 
The  best  way  of  ascertaining  whether  or  not  this 
>^        condition   exists   is   to  illuminate  the  slit  with 


monochromatic  light,  such  as  that  produced  by  incandescent 
sodium  vapor,  and  then  to  see  whether  or  not  a  sharply  denned 
image  of  the  slit  is  formed  in  the  eyepiece  of  the  telescope. 
190.  Normal  and  prismatic  spectra.  In  Chapter  XXII  it  was 
shown  that  the  spectrum  produced  by  a  grating  is  one  in  which 
the  angular  separation  of  any  two  colors  is  directly  proportional 
to  the  difference  in  wave  length  between  these  colors,  provided  6 
is  small  (see  p.  2  64)  ;  for  then 


This  means  that  two  photographs  made  with  two  different  grat- 
ings, but  reduced  to  the  same  size,  are  identical  in  the  arrangement 
and  proportion  of  their  colors.  Because  of  the  ease  of  comparison 
resulting  from  this  fact  the  grating  spectrum  has  been  adopted  as 
the  standard,  or  normal,  spectrum. 


DISPEKSION  AND  SPECTKA 


305 


FIG.  214 


Iii  general,  the  spreading  of  the  different  colors  produced  by  the 
passage  of  light  through  a  prism  is  not  at  all  proportional  to  the 
difference  in  wave  lengths,  nor  indeed  does  a  substance  which 
produces  a  large  mean  refraction  always  produce  a  correspondingly 
large  spreading  or  dispersion  of  any  two  colors.  In  other  words, 
dispersion  is  not  proportional  to 
refraction,  although  Newton  sup- 
posed, from  his  early  investigation 
of  the  subject,  that  such  propor- 
tionality existed.  In  general,  pris- 
matic spectra  differ  from  normal 
spectra  in  that  the  reds  and  yellows 
are  relatively  little  separated,  while 
the  blues  and  violets  are  abnormally  spread  out.  If,  then,  a 
photograph  of  a  spectrum,  produced  by  a  prism  is  made  to  the 
same  scale  as  one  produced  by  a  grating,  the  different  colors 
will  not  occupy  at  all  the  same  positions  or  the  same  relative 
spaces.  Nor,  indeed,  are  the  spectra  of  prisms  made  of  different 
materials  found  to  agree  with  one  another,  the  red  and  yellow,  for 
example,  suffering  a  larger  relative  separation  in  one  case  than 
in  another.  It  is  on  account  of  this  so-called  irrationality  of 
prismatic  dispersion  that  it  is  possible  to  construct  direct-vision 
spectroscopes.  These  instruments  consist  of  prisms  so  combined 
as  to  produce  dispersion  without  producing  any  mean  deviation  of 
the  beam  (see  Fig.  214).  It  is  also  on  account  of  the  irrationality 
of  dispersion  that  it  is  possible  to  produce  so-called  achromatic 
lenses.  These  will  be  considered  in  the  following  section. 

191.  Chromatic  aberration  and  the  achromatic  lens.    The  fact 
that  a  glass  lens  produces  dispersion  is  responsible  for  a  phenome- 
non observable  with  all  simple 
lenses  and  known  as  chromatic 
aberration.    When  white  light 
falls  upon  such  a  lens,  since 
FlG-  215  the  violet  waves  are  refracted 

more  than  the  red  ones,  the  focus  for  the  violet  waves  must  obviously 
be  closer  to  the  lens  than  is  that  for  the  red.  If  v  and  r  (Fig.  215) 
represent  these  two  foci  respectively,  then  the  foci  of  the  colors 


v   r 


306  ELECTRICITY,  SOUND,  AND  LIGHT 

intermediate  between  violet  and  red  will  obviously  occupy  positions 
intermediate  between  v  and  r.  It  is  because  of  this  phenomenon 
that  the  image  formed  by  a  simple  lens  is  in  general  indistinct 
and  fringed  with  color.  If  a  card  is  held  nearer  to  the  lens  than 
the  mean  focal  plane,  the  outer  edge  of  the  image  is  fringed  with 
red,  since  the  red  rays,  being  least  refracted,  are  here  on  the  outside, 
as  shown  in  the  figure.  If  the  card  is  moved  to  a  position  just 
beyond  the  mean  focal  plane,  the  image  is  fringed  with  violet, 
since  the  violet  rays,  after  crossing  in  the  focal  plane,  are  here  on 
the  outside. 

The  problem  of  eliminating  chromatic  aberration  is  obviously 
the  inverse  of  the  problem  of  constructing  a  direct-vision  spectro- 
scope, for  in  the  latter  it  is  necessary  to  produce  dispersion  with- 
out producing  mean  deviation,  while  in  the  former  it  is  necessary 
to  produce  refraction  without  producing  dispersion. 
The  problem  is  solved  by  combining  into  one  lens 
a  convex  lens  of  crown  glass  and  a  concave  lens  of 
flint  glass  in   the  manner  shown  in  Figure  216. 


FIG  216         The  nmt-glass  lens  then  overcomes  practically  all 

the  dispersion  produced   by  the  crown-glass   lens, 

without,  however,   overcoming    the    refraction.     Such  lenses  are 

called  achromatic  lenses.    They  are  used  in  the  construction  of 

all  high-grade  optical  instruments. 

192.  Continuous  and  discontinuous  spectra.  In  general  when 
a  pure  spectrum  is  formed  of  the  light  emitted  by  an  incandescent 
solid  or  liquid,  it  is  found  that  there  are  no  breaks  whatever  in 
the  band  of  color  which  constitutes  the  spectrum,  no  matter  how 
narrow  the  slit  may  be.  Looked  at  from  Newton's  standpoint  this 
means  that  the  light  emitted  by  a  white-hot  solid  or  liquid  con- 
tains every  conceivable  wave  length  between  the  longest  red  and 
the  shortest  violet.  All  that  we  are  able  to  assert  with  certainty, 
however,  is  that  the  light  which  has  passed  through  the  prism 
contains  all  conceivable  wave  lengths  within  these  limits. 

When,  however,  a  gas  or  vapor  is  brought  to  incandescence,  its 
pure  spectrum  is  found  to  consist  of  a  number  of  separate  images 
of  the  slit,  each,  of  course,  in  its  own  color.  This  means  that  an 
incandescent  gas  or  vapor  emits  only  radiations  of  certain  definite 


DISPERSION  AND  SPECTRA  307 

wave  lengths.  Spectra  of  this  type  are  called  bright-line  spectra. 
They  are  produced  by  vaporizing  metallic  salts  in  a  hot  flame,  like 
that  of  the  Bunsen  burner,  or  by  sending  electrical  discharges 
through  tubes  containing,  in  rarefied  form,  the  gases  to  be  ex- 
amined. The  characteristic  spectrum  of  sodium,  for  example,  is 
usually  formed  by  placing  common  salt  or  some  other  compound 
containing  sodium,  for  example  an  ordinary  glass  rod,  in  a  Bunsen 
flame.  It  consists  of  two  bright  yellow  lines  very  close  together. 
In  ordinary  spectroscopic  work  it  is  seen  simply  as  a  single  line, 
but  if  the  slit  of  the  spectrometer  is  made  exceedingly  narrow,  the 
original  broad  image  of  a  broad  slit  is  found  to  separate  into  two 
narrow  images  of  the  very  narrow  slit,  thus  showing  that  the 
images  were  not  originally  distinguished  as  two  merely  because 
they  overlapped.  The  spectra  of  other  gases  and  vapors  are  not 
so  simple  as  that  of  sodium,  and  the  lines  are  in  general  scattered 
through  the  whole  range  of  visible  wave  lengths.  The  fact  that 
the  spectrum  of  an  ordinary  gas  flame  is  of  the  continuous  rather 
than  of  the  bright-line  type  is  due  to  the  fact  that  the  incan- 
descent body  in  a  gas  flame  is  not  a  gas  at  all,  but  is  rather  solid 
carbon  particles  suspended  in  a  nearly  colorless  flame  like  that  of 
the  Bunsen  burner. 

193.  Absorption  spectra.  In  addition  to  the  bright-line  spectra 
discussed  above  there  is  another  type  of  discontinuous  spectrum, 
namely  the  so-called  dark-line  or  absorption  spectrum.  When  a 
pure  spectrum  is  formed  by  the  light  from  the  sun  after  it  has 
passed  through  a  sufficiently  narrow  slit,  it  is  found,  upon  exami- 
nation, to  consist  of  a  continuous  spectrum  crossed  by  a  large  num- 
ber of  fine  dark  lines.  These  lines  were  first  noticed  by  Wollaston 
in  1802,  but  were  afterwards  rediscovered  in  1814  and  investigated 
by  Fraunhofer,  who  located  about  700  of  them,  and  after  whom 
they  were  named.  The  existence  of  these  lines  in  the  solar  spec- 
trum shows  clearly  that  certain  wave  lengths  are  either  absent 
from  the  sunlight  which  has  passed  through  the  prism,  or,  if  not 
entirely  absent,  are  at  least  much  weaker  than  are  their  neighbors. 
When  the  solar  spectrum  is  compared  with  a  sodium  spectrum 
formed  by  the  same  spectrometer,  it  is  found  that  two  dark  lines 
in  the  former  are  exactly  identical  in  position  with  the  two  sodium 


308  ELECTRICITY,  SOUND,  AND  LIGHT 

lines.  A  similar  comparison  of  the  spectra  of  other  elements  with 
the  solar  spectrum  has  resulted  in  the  identification  in  position  of 
many  of  the  dark  lines  of  the  latter  with  the  bright  lines  of  the 
former.  An  explanation  of  this  phenomenon  is  suggested  by  the 
following  experiment. 

When  a  pure  solar  spectrum  is  formed  by  means  of  any  instru- 
ment, it  is  found  that  if  the  intensely  yellow  flame  which  is  pro- 
duced by  burning  metallic  sodium  is  placed  anywhere  in  the  path 
of  the  beam  of  sunlight  which  falls  upon  the  slit  of  the  spectrom- 
eter, the  dark  lines  of  the  solar  spectrum  which  correspond  in 
position  with  the  bright  lines  of  the  sodium  spectrum  are  intensified, 
that  is,  they  are  very  much  darker  than  before.  This  seems  to 
show  clearly  that  the  two  prominent  lines  in  the  yellow  part  of 
the  solar  spectrum  are  due  in  some  way  to  sodium  vapor  through 
which  the  sunlight  has  somewhere  passed  on  its  way  to  the  spec- 
trometer, since  making  it  pass  through  more  sodium  vapor  increases 
the  prominence  of  these  lines.  Now  we  know  that  whenever  the 
waves  from  a  sounding  tuning  fork  fall  upon  another  fork  of 
exactly  the  same  pitch,  the  latter  is  set  into  sympathetic  vibrations  ; 
in  other  words,  the  second  fork  absorbs  the  vibrations  emitted  by 
the  first.  We  know  further  that  this  phenomenon  of  sympathetic 
vibrations  cannot  be  produced  unless  the  two  forks  have  precisely 
the  same  natural  periods.  It  is  customary  to  assume,  therefore,  in 
explanation  of  the  two  dark  lines  in  the  yellow  portion  of  the 
solar  spectrum,  that  the  extremely  intense  radiation  of  all  wave 
lengths,  due  to  the  extremely  hot  solid  nucleus  of  the  sun,  has  had 
some  of  its  wave  lengths  weakened  by  absorption  as  it  has  passed 
through  the  cooler  sodium  vapor  in  the  gaseous  envelope  (the 
chromosphere)  of  the  sun,  but  that  the  only  wave  lengths  so 
weakened  are  those  which  correspond  to  the  exact  periods  of 
vibration  which  the  absorbing  vapor  itself  is  capable  of  emitting. 
A  similar  explanation  holds  for  the  other  Fraunhofer  lines.  These 
lines  are  not  then,  in  general,  devoid  of  light,  but  merely  appear 
dark  in  the  solar  spectrum  because  of  the  very  much  greater 
intensity  of  the  light  of  adjacent  wave  lengths  which  have  not 
been  weakened  by  absorption.  Thus  the  same  sodium  vapor  which, 
when  viewed  by  itself  in  the  spectroscope,  appears  as  two  bright 


DISPEESION  AND   SPECTKA 


309 


lines,  appears  as  two  dark  lines  when  viewed  against  the  brilliant 
background  of  the  solar  spectrum.  In  other  words,  the  darkness 
is  merely  a  matter  of  contrast.  Figure  217  shows  the  location  of 
the  main  lines  of  the  normal  (see  p.  263)  solar  spectrum.  The 
corresponding  wave  lengths  are  given  in.  so-called  Angstrom  units 
or  10~10  meters.  Some  of  these  lines,  such  as  A  and  B,  are  known 
to  be  due  to  absorption  which  takes  place  in  the  earth's  atmosphere. 
A  and  B  are  in  fact  due  to  the  oxygen  of  our  air.  The  group  b  is 
due  to  magnesium  vapor  in  the  sun.  C,  F,  and  h  are  due  to  hydro- 
gen in  the  sun.  Those  of  the  lines  which  are  due  to  absorption 
in  the  sun  are  distinguished  from  those  due  to  absorption  in  the 
earth's  atmosphere  by  the  fact  that  when  the  light  from  that  edge 


Red                               Orange           Yellow            Green    Blue  Green               Violet 

FIG.  217 

of  the  sun  which,  in  view  of  the  sun's  rotation,  is  moving  toward 
the  earth,  is  thrown  into  the  slit  of  a  spectroscope,  the  lines  which 
have  their  origin  in  the  sun  are  slightly  displaced  toward  the  violet 
end  of  the  spectrum  in  accordance  with  the  principle,  known  as 
Doppler's  principle,  that  when  a  vibrating  body  moves  toward  an 
observer  the  wave  length  is  shortened.  Lines  due  to  our  atmosphere 
obviously  could  not  show  such  displacement. 

194.  The  reversal  of  the  sodium  lines.  The  phenomenon  of  the 
absorption  of  the  light  emitted  by  sodium  vapor  when  this  light 
is  passed  through  cooler  layers  of  sodium  vapor  may  be  shown  as 
follows.  If  the  light  produced  by  burning  metallic  sodium  in  the 
Bunseii  flame  is  observed  through  the  spectrometer,  the  spectrum 
will  at  first  consist  simply  of  the  two  brilliant  sodium  lines,  but 
as  the  burning  continues  a  point  is  reached  at  which  the  hot 
sodium  vapor  in  the  interior  of  the  flame  has  become  surrounded 


310  ELECTRICITY,  SOUND,  AND  LIGHT 

by  dense  masses  of  its  own  vapor  at  a  lower  temperature.  At  this 
moment  the  absorption  occurs  and  the  centers  of  the  two  lines 
suddenly  turn  black.  This  phenomenon  can  be  observed  in  any 
dense  incandescent  vapor.  It  is  called  the  phenomenon  of  reversal. 

195.  Theory  of  bright-line  and  continuous  spectra.  The  fact 
that  gases  and  vapors  give  forth  only  certain  definite  wave  lengths 
is  easily  explained  from  the  standpoint  of  the  kinetic  theory  of 
matter.  For  since,  according  to  this  theory,  the  molecules  of  ordi- 
nary gases  are  for  the  most  part  outside  the  range  of  one  another's 
influence,  it  is  to  be  expected  that  when  their  constituent  parts, 
for  example,  their  electrons,  are  once  set.  into  vibration  by  any 
cause,  they  will  continue  to  vibrate  in  their  natural  periods  quite 
undisturbed  during  the  whole  interval  between  two  impacts. 
Hence  these  vibrating  electrons  will  send  out  relatively  long 
trains  of  perfectly  definite  wave  lengths.  Furthermore,  if  the  atom 
is  a  very  complex  structure,  it  is  entirely  possible  that  a  given 
atom  might  send  forth  a  large  number  of  different  wave  lengths. 
This  picture  of  the  mechanism  of  light  emission  by  incandescent 
gases  requires,  indeed,  a  very  great  complexity  in  some  atoms,  such, 
for  example,  as  that  of  iron,  the  spectrum  of  which  contains  sev- 
eral thousand  different  lines. 

The  explanation  of  the  continuous  spectrum  of  incandescent 
solids  offers  greater  difficulty.  A  theory  which  was  in  vogue  up 
to  about  1895  was  developed  in  view  of  the  three  following  facts. 

(1)  If  the  density  of  a  gas  is  increased,  its  spectral  lines  grow 
broader. 

(2)  When  liquid  solutions  show  absorption  spectra  the  absorption 
bands  are  never  fine  dark  lines,  as  in  the  case  of  gases,  but  are 
in  general  broad  bands. 

(3)  Theory  shows  that  although  a  body  which  vibrates  without 
damping  must  have  a  perfectly  definite  and  unchanging  period, 
and  must  therefore  emit  a  homogeneous  train  of  waves,  a  vibration 
which   occurs   with   considerable   damping   is   one   of   constantly 
changing  period,  the  limits  of  which  are  larger  and  larger  the 
greater  the  damping.    If,  then,  we  assume  that  the  closely  packed 
molecules  of  liquids  and  gases  are  capable,  on  account  of  their 
mutual  influences,  of  producing  only  strongly  damped  vibrations, 


DISPERSION  AND   SPECTRA  311 

we  can  account  for  the  fact  that  matter  in  the  rarefied  condition 
shows  narrow  absorption  bands  and  emits  bright-line  spectra,  while 
in  the  dense  condition  it  has  broad  absorption  bands  and  emits, 
with  increasing  density,  broader  and  broader  lines  which  finally 
run  together  into  the  continuous  spectrum. 

It  has  been  pointed  out,  however,  especially  by  Gouy  in  France 
and  Rayleigh  in  England,  that  it  is  not  necessary  to  assume  any 
periodicity  at  all  in  the  source  which  emits  white  light.  For  it  can 
be  shown  by  mathematical  analysis  that  it  is  possible  to  resolve 
an  irregular  jumble  of  pulses,  such  as  might  be  communicated  to 
the  ether  by  atomic  shocks,  into  such  a  number  of  homogeneous 
vibrations,  having  periods  which  lie  very  close  together,  as  is 
found  in  the  continuous  spectrum.  Indeed,  if  we  assume  that  the 
particles  of  an  incandescent  body  do  vibrate  in  an  infinite  num- 
ber of  different  periods,  it  is  clear  that,  since  these  vibrations  must 
all  be  transmitted  simultaneously  to  the  eye  by  the  same  ether, 
the  resultant  disturbance  of  any  particular  point  or  particle  of  the 
ether  would  be  very  irregular,  and  even  according  to  Newton's  view 
point  this  irregular  disturbance  must  be  resolved  by  the  prism 
or  grating  into  the  extremely  close  series  of  regular  wave  lengths 
which  the  continuous  spectrum  shows.  It  is  therefore  not  neces- 
sary to  assume  that  white  light  ever  consisted  of  anything  but  the 
jumble  of  irregular  pulses  which,  in  any  case,  the  prism  or  grating 
is  obliged  to  resolve.  It  is  then  very  easy  to  see  how  a  body  like 
a  solid  or  liquid,  in  which  the  molecules  make  extremely  short 
excursions  between  impacts,  might  emit  such  a  jumble  of  ether 
pulses  as  this  theory  requires  for  the  constitution  of  white  light. 

196.  Spectroscopic  analysis.  Since  a  given  substance  in  a 
gaseous  condition  always  has  a  characteristic  spectrum,  and  since, 
furthermore,  the  spectrum  of  an  elementary  substance  like  hydro- 
gen is  in  general  found  to  appear  in  the  spectrum  of  any  compound 
containing  hydrogen,  it  will  be  seen  that  the  observation  of  the 
character  of  the  spectrum  of  a  substance  of  unknown  composition 
furnishes  a  very  satisfactory  method  of  testing  for  the  presence 
of  certain  substances  in  the  compound.  Since  gases  alone  have 
characteristic  spectra,  the  method  of  spectroscopic  analysis  is 
obviously  limited  to  the  observation  of  the  spectra  of  vaporized 


312 


ELECTRICITY,  SOUND,  AND  LIGHT 


substances.    There  are,  in  general,  three  ways  in  which  gaseous 
spectra  are  compared. 

(1)  The  first  is  represented  in  Figure  218.  The  image  of  an 
illuminated  scale  L  is  formed  in  the  focal  plane  of  the  eyepiece 
0  of  the  telescope  T  by  means  of  a  reflection  from  one  face  of 
the  prism  P.  At  the  same  time  some  substance,  for  example, 


FIG.  218 

lithium  chloride,  is  vaporized  in  the  flame  $,  and  the  character- 
istic spectrum  of  lithium  is  therefore  also  formed  in  the  focal 
plane  of  the  eyepiece  0.  The  exact  positions  of  the  lithium  lines 
on  the  scale  are  recorded.  Some  unknown  compound  which  is  to 
be  tested  for  the  presence  of  lithium  is  then  vaporized  in  the 

flame  $,  or  in  a  spark  tube  which 
replaces  $,  and  the  presence  of  lines 
on  the  illuminated  scale  in  exactly 
_  the  position   just  occupied   by  the 

lithium  lines  is  looked  for  in  the 
spectrum  of  this  compound. 

(2)  In  the  second  method  the  slit 
of  the  spectroscope  is  covered  through  half  its  length  by  means 
of  a  small  total-reflecting  prism  P  (Fig.  219).  Light  from  the 


'/I 


t 


FIG.  219 


DISPERSION  AND  SPECTRA 


313 


compound  is  then  caused  to  enter  the  upper  half  of  the  slit  from 

the  flame  a,  while  light  from  a  lithium  flame  b,  for  example,  is 

caused  to  enter  the  spectroscope  through  the  lower 

half  of  the  slit  by  means  of  total  reflection  within 

the  prism  P.    If,  then,  the   characteristic  lines   of 

lithium  which  appear  in  the  upper  half  of  the  field 

of  view  of  the  telescope  T  are  found  to  continue 

clear  across  the  field  of  view,  then  lithium  must  be 

in   the   compound   which   is   being   vaporized   at  a, 

since  the  spectrum  of  this  compound  occupies  only 

the  lower  half  of  the  field  of  view.    This  method 

evidently  renders  the  scale  L  unnecessary. 

(3)  The  third,  and  perhaps  the  most  satisfactory 
method,  consists  in  replacing  the  telescope  T  by  a 
photographic  camera  and  taking  a  photograph  of 
the  unknown  substance  which  is  being  vaporized 
at  a  with  the  photographic  plate  partially  covered, 
and  then  replacing  the  unknown  substance  at  a 
by  the  substance  for  which  the  test  is  being  made, 
and,  without  altering  at  all  the  position  of  the 
plate,  taking  another  photograph  when  only  the 
.previously  covered  portion  of  the  plate  is  exposed. 
It  is  then  only  necessary  to  see  whether  the  char- 
acteristic lines  obtained  by  the  last  exposure  coin- 
cide with  those  obtained  in  the  first  exposure. 
Figure  220  is  a  copy  of  a  photograph  so  taken, 
the  upper  and  lower  portions  representing  the 
bright-line  spectrum  of  iron,  and  the  middle  por- 
tion the  dark-line  spectrum  of  the  sun.  It  is 
obvious  that  iron  is  present  in  the  sun.  Some  of 
the  other  substances  which  have  been  identified  in 
this  way  in  the  solar  spectrum  are  calcium,  oxygen, 
hydrogen,  aluminum,  nickel,  magnesium,  cobalt,  sili- 
con, carbon,  copper,  zinc,  cadmium,  silver,  tin,  and 
lead.  The  lines  characteristic  of  the  element  now 
known  as  helium  were  observed  in  the  sun  before  the  element 
was  known  to  exist  on  the  earth. 


FIG.  220 


314  ELECTRICITY,  SOUND,  AND  LIGHT 

EXPERIMENT  26 

(A)  Object.    To  become  familiar  with  the  spectra  of  different  substances. 
Directions.    Using  a  spectroscope  either  of  the  form  shown  in  Figure 

194  or  Figure  218,  see  that  the  prism  is  in  approximately  the  position 
of  minimum  deviation  for  sodium  light  (see  sect.  175,  p.  280).  Using  a 
slit  about  half  a  millimeter  wide,  focus  the  telescope  until  the  image  of 
the  slit  is  in  sharp  focus.  Then  replace  the  sodium  light  by  a  white  light 
and  make  a  rough  chart,  similar  in  form  to  that  shown  in  Figure  222,  of  the 
distribution  of  the  light  in  the  spectrum,  indicating  on  the  chart  by  brackets 
or  otherwise  what  portions  of  the  total  length  of  the  spectrum  are  occupied 
by  the  red,  the  yellow,  the  green,  the  blue,  and  the  violet  respectively. 

Replace  the  white  light  by  sodium  light,  and  reduce  the  width  of  the 
slit  until  the  yellow  sodium  line  is  seen  to  be  in  reality  two  distinct  lines 
very  close  together.  If  it  does  not  appear  as  such  at  first,  make  the  slit 
still  narrower  and  focus  the  eyepiece  more  sharply.  If  it  is  not  even  then 
double,  it  is  probable  that  the  spectroscope  has  not  a  sufficiently  high 
resolving  power.  Now  make  a  chart  of  the  sodium  spectrum; 
that  is,  draw  two  fine  lines,  or  one,  if  but  one  is  seen,  beneath  the 
central  portion  of  the  region  marked  "  yellow  "  in  the  chart  above 
and  label  it  "  sodium."  If  you  are  using  a  spectroscope  of  the  kind 
shown  in  Figure  218,  indicate  the  exact  numerical  position  of  the 
sodium  lines  upon  the  scale. 

Using  a  relatively  wide  slit  (.5  mm.),  introduce  successively  with 
different  platinum  wires  into  the  flame  of  a  Bunsen  burner  S 
(Fig. '218)  the  chlorides  of  lithium,  strontium,  calcium,  barium, 
and  potassium,  and  make  a  chart  of  the  spectrum  of  each.  In 
the  case  of  strontium  notice  particularly  the  isolated  line  in  the 
blue;  in  the  case  of  potassium,  the  line  in  the  extreme  red. 
Some  of  these  lines  are  persistent,  wrhile  others  appear  only  for 
an  instant  after  the  salt  is  introduced  into  the  flame. 

In  a  similar  way  make  charts  of  the  spectra  of  nitrogen,  mer- 
cury, and  hydrogen,  obtaining  these  spectra  by  means  of  an  induc- 
FIG  221  tion-coil  discharge  through  a  vacuum  tube  (see  Fig.  221)  placed 
in  front  of  the  slit. 

(B)  Object.    To  analyze  a  mixture  for  the  presence  of  various  elements. 
Directions.    By  either  of  methods  (1)  or  (2)  in  section  196  analyze  an 

unknown  mixture  furnished  by  the  instructor. 

(C)  Object.    To  observe  the  phenomenon  of  the  reversal  of  the  spectral 
lines  of  sodium. 

Directions.  Use  again,  if  possible,  a  sufficiently  narrow  slit  to  bring  out 
the  sodium  spectrum  as  a  close  double  line.  Burn  a  piece  of  metallic  sodium 
about  twice  as  large  as  a  pea  in  a  Bunsen  flame  placed  a  foot  or  so  in  front 
of  the  slit,  and  observe  the  reversal  of  the  sodium  lines. 


DISPERSION  AND  SPECTRA 


315 


(D)  Object.    To  show  the  existence  of  sodium  and  hydrogen  Jn  the  sun 
and  to  compare  a  prismatic  with  a  normal  solar  spectrum. 

Directions.  With  a  lens  of  about  the  same  focal  length  as  that  of  the 
collimator  throw  an  image  of  the  sun  upon  the  slit  of  the  spectroscope. 
Make  this  slit  very  narrow,  focus  the  eyepiece  until  the  Fraunhofer  lines 
are  very  distinct,  then  identify  as  many  as 
possible  of  the  lines  A,  a,  B,  C,  D,  E,  b,  F, 
and  G,  of  Figure  222,  which  shows  a  pris- 
matic spectrum.  Xote  that  the  distances 
apart  of  A  and  D,  D  and  F,  and  F  and  G 
are  approximately  the  same,  and  also  that 
the  distance  from  B  to  D  is  about  the  same 
as  that  from  D  to  b.  Place  a  sodium  flame 
before  the  slit  and  note  that  when  the  sun- 
light is  cut  off  the  yellow  sodium  lines  ap- 
pear in  the  exact  position  of  the  D  lines.  In 
the  same  way  show  that  C  and  F  are  hydro- 
gen lines. 

Replace  the  prism  by  a  reflection  grating, 
or  take  another  instrument  in  which  a  grat- 
ing is  in  place,  and  set  this  grating  in  such  a 
position  that  light  coming  through  the  slit 
falls  upon  the  grating  at  an  angle  of  inci- 
dence not  exceeding  45  degrees.  Turn  the 
telescope  so  as  to  take  in  the  light  reflected 
from  the  grating  face  in  accordance  with  the 
law,  "  angle  of  incidence  equals  angle  of  re- 
flection." This  light  should  produce  in  the 
field  of  the  telescope  an  uncolored  image  of 
the  slit.  This  image  corresponds  to  the  cen- 
tral uncolored  image  produced  by  the  trans- 
mission grating  of  Experiment  22.  Focusing, 
as  above,  the  sun's  rays  upon  a  very  narrow 
slit,  adjust  the  leveling  screws  of  the  grating, 
or  grating  table,  until  the  uncolored  image 
of  this  slit  is  in  the  middle  of  the  field  of 
view.  Then  turn  the  telescope  and  observe 
the  spectra  of  the  first  and  second  orders  on 
either  side  of  the  central  image.  Observe  and  record  the  distances  apart 
in  this  normal  spectrum  of  A,  D,  F,  and  G,  and  of  B,  Z>,  and  b.  Com- 
pare these  distances  with  those  shown  in  Figure  217.  See  if  the  distance 
apart  of  the  D  lines  of  the  sun's  spectrum,  or  the  bright  lines  of  the  sodium 
spectrum,  is  not  much  greater  in  the  spectrum  of  the  second  order  than  it 
is  in  that  of  the  first  (see  sect.  162,  p.  264). 


CHAPTEE   XXYII 


POLARIZED   LIGHT 

197.  Polarization  by  reflection.  All  of  the  phenomena  of  light 
which  have  been  thus  far  studied  have  been  found  to  be  expli- 
cable upon  the  basis  of  the  same  wave  theory  whicli  applies  to  the 
phenomena  of  sound.  In  other  words,  so  far  as  the  fundamental 
facts  of  reflection,  refraction,  diffraction,  emission,  and  absorption 
are  concerned,  sound  and  light  are  identical  in  all  respects  except 
in  the  lengths  of  their  waves  and  in  the  nature  of  the  media 
which  act  as  their  carriers. 

There  is,  however,  a  class  of  phenomena,  known  as  the  phenom- 
ena of.  polarization,. which  differentiate  light  completely  from  sound, 
and  show  that  light  waves  are  not  compressional  waves  at  all 
as  are  sound  waves,  but  are  instead  transverse  waves  similar  to 

those  which  elastic  solids  are  able  to 


propagate  by  virtue  of  their  rigidity. 
These  phenomena  are  so  far  removed 
from  ordinary  observation  that  they 
will  be  here  presented  in  connection 
with  a  series  of  qualitative  experi- 
ments. The  facts  presented  in  the 
first  experiment  were  discovered  in 
1810  by  the  French  physicist  Malus 
(1775-1812). 


Experiment  1.  Set  the  plane  glass  re- 
flector m  of  the  so-called  Norrenberg  po- 
lariscope  of  Figure  223  so  that  its  plane 
makes  an  angle  of  about  33°  with  the 
vertical.  Adjust  the  position  of  a  hori- 
zontal slit  s  (about  5  mm.  wide)  and  a 
sodium  flame  /  so  that  when  you  remove  the  black  glass  mirror  m'  and 
look  vertically  down  upon  the  middle  of  m  you  see  a  portion  of  the  flame. 

316 


FIG.  223 


POLARIZED  LIGHT 


31' 


In  this  experiment  the  mirror  n  may  be  covered  with  a  piece  of  black 
paper.  Place  m'  in  position  and  turn  it  so  that  it  is  exactly  parallel  to  m, 
that  is,  so  that  its  plane  also  makes  an  angle  of  33°  with  the  vertical. 
Place  the  eye  at  E  in  such  a  position  that  when  you  look  at  the  middle 
of  in  you  see  the  twice-reflected  image  of  the  sodium  flame.  Then  rotate 
in'  in  its  frame  about  a  vertical  axis  and  observe  the  image  of  the  flame 
as  you  do  so.  When  you  have  turned  m'  through  90°,  that  is,  into  the 
position  shown  in  Figure  223,  2,  the  image  of  the  flame  will  have  com- 
pletely disappeared. 

The  experiment  shows  that  light  waves  cannot  be  longitudinal, 
for  if  the  particles  of  the  medium  which  transmits  the  light  from 
m  to  m1  vibrated  in  the  direction  of  propagation  of  the  light, 
then  the  conditions  of  symmetry  would  demand  that  the  wave  be 
reflected  in  precisely  the  same 
way  after  m'  has  been  rotated 
through  90°  as  before.  But 
if  light  consists  of  waves  in 
which  the  direction  of  vibra- 
tion of  the  particles  of  the 
medium  is  always  transverse 
to  the  direction  of  propaga- 
tion of  the  waves,  and  if,  in 
a  very  short  interval  of  time, 
the  vibrating  particles  which 
give  rise  to  light  waves  change 

their  direction  of  vibration  many  times,  then  the  above  phenomena 
can  be  very  easily  understood.  For  suppose  that  in  Figure  224 
dbt  ccl,  ef,  gli,  etc.,  represent  successive  directions  of  vibration  of 
the  particles  of  the  medium  across  the  path  of  the  ray  sm  (Figs. 
223  and  224).  All  of  these  vibrations  can  be  resolved  into  two 
component  vibrations,  the  one  perpendicular  to  the  plane  of  inci- 
dence smm'j  that  is,  at  right  angles  to  the  plane  of  the  page,  and 
represented  by  the  dots  in  the  line  sm,  and  the  other  in  the  plane 
of  incidence  and  represented  by  the  straight  lines  drawn  across 
the  path  of  the  ray  sm. 

Now  when  the  ray  sm  strikes  the  mirror  it  is  clear  that  the 
general  law  of  reflection,  namely  angle  of  incidence  equals  angle 
of  reflection,  requires  that  there  be  some'  angle  of  incidence  such 


FIG.  224 


318  ELECTRICITY,  SOUND,  AND  LIGHT 

that  the  refracted  ray  mr  and  the  reflected  ray  mm'  are  at  right 
angles  to  each  other.  But  when  this  is  the  case  that  component 
vibration  of  the  refracted  ray  which  lies  in  the  plane  of  incidence 
coincides  with  what  should  be  the  direction  of  the  reflected  ray, 
namely  the  line  mm'-,  hence  this  component  vibration  obviously 
has  no  component  which  is  perpendicular  to  mm'.  But  if  light 
vibrations  are  always  perpendicular  to  the  direction  of  propaga- 
tion of  the  light,  this  means  that  there  should  be  one  angle 
of  incidence  for  which  no  part  of  the  component  vibration 
of  the  original  light  which  is  in  the  plane  of  incidence  can  be 
reflected. 

Considerations  of  symmetry  require,  however,  that  that  compo- 
nent vibration  of  the  ray  sm  which  is  perpendicular  to  the  plane  of 
incidence,  and  represented  by  the  dots  in  the  figure,  should  be 
reflected  at  all  angles  of  incidence.  Now  this  is  precisely  what 
experiment  shows  to  be  the  case.  At  the  angle  of  incidence  of  the 
ray  sm  for  wliich  the  angle  rmm'  is  a  right  angle,  the  reflected  ray 
mm'  consists  only  of  vibrations  which  are  perpendicular  to  the 
plane  of  incidence.  The  ray  mm'  is  said  to  be  a  ray  of  plane 
polarized  light,  and  the  angle  of  incidence  at  which  the  ray  sm 
must  fall  upon  the  mirror  in  order  that  the  reflected  ray  mm'  may 
consist  only  of  vibrations  in  this  one  plane  is  called  the  polariz- 
ing angle.  That  this  angle  is  always  the  angle  for  which  the 
reflected  and  refracted  are  at  right  angles  was  discovered  in  1815 
by  Sir  David  Brewster  (1781-1868),  and  is  known  as  Brewster's 
law.  It  may  easily  be  shown  that  another  form  of  statement  of 
the  same  law  is  the  following:  the  angle  of  complete  polariza- 
tion is  the  angle  the  tangent  of  which  is  the  index  of  refraction 
of  the  reflecting  substance.  This  is  the  form  in  which  Brewster 
announced  Ms  law.*  That  the  two  forms  of  statement  repre- 
sent one  and  the  same  physical  relation  may  be  seen  from  the 
following : 

If  the  angle  cod  (Fig.  225)  is  90°,  then  we  have  90°  —  ^+90° 
-  r  =  90°,  or  i  +  r  =  90°;  hence  sin  r  =  cos  i ;  hence  the  index 
n(=  sin  fc'/sin  r)  may  be  written  in  the  form 

sin*  .  .  _! 

n  =  —  —  —  tan  ^)      or      ^  =  tan     n.  (1) 

COS2 


POLAEIZED  LIGHT 


319 


FIG.  225 


The  reason  that  we  originally  set  the  mirror  m  so  as  to  make 
an  angle  of  33°  with  the  vertical  was  that  the  index  of  refraction 
of  crown  glass  is  about  1.55,  and  the  angle  the  tangent  of  which 
is  1.55  is  57°.  In  order  that  the  angle  of  incidence  might  be 
57°  it  was  necessary  to 
make  the  angle  between 
the  plane  of  the  mirror 
and  the  vertical  33°. 

It  will  now  be  obvious 
why  we  obtained  no  re- 
flected light  at  all  from 
mf  when  we  had  rotated 
it  from  position  1  to  po- 
sition 2  (Fig.  223).  For 
in  this  latter  position  mr  bore  precisely  the  same  relation  to  the 
vibration  of  the  ray  mm'  as  did  the  mirror  m  to  the  component  of 
sm  which  was  vibrating  in  the  plane  of  incidence  smm'. 

Experiment  2.  Kow  set  m  and  m'  again  so  that  there  is  no  light  from/ 
reflected  at  mf  and  then  rotate  mf  about  a  horizontal  axis,  observing  the 
middle  of  in'  all  the  while.  You  will  find  that  there  is  always  some  of  the 
light  ray  mm'  reflected  from  m'  except  when  m'  is  set  exactly  at  the  polar- 
izing angle.  The  amount  of  the  light  thus  reflected 
will  be  found  to  increase  rapidly  as  the  position  of 
the  mirror  departs  in  either  direction  from  the  polar- 
izing angle. 

Experiment  3.  Replace  the  glass  mirror  m'  by  a 
pile  of  about  fifteen  thin  glass  plates  set  at  the  polar- 
izing angle  (see  Fig.  226),  and  then  observe  not  only, 
as  above,  the  reflected  ray  e,  but  also  the  ray  e'  trans- 
mitted by  the  plates  as  the  pile  is  turned  about  a 
vertical  axis.  You  will  find  that  when  the  pile  of 
plates  is  in  position  2  (Fig.  223),  that  is,  in  the  posi- 
tion such  that  the  reflected  ray  disappears,  the  trans- 
mitted ray  is  of  maximum  brightness,  and  when  the 
plates  are  rotated  into  position  1  (Fig.  223),  that  is, 
into  a  position  such  that  the  reflected  ray  is  of  maximum  brightness,  the 
transmitted  ray  has  almost  entirely  disappeared. 

In  explanation  of  these  effects  consider  that  an  incident  beam 
sm  (Fig.  227)  of  ordinary  light  is  resolved  into  two  components, 


FIG.  226 


320 


ELECTRICITY,  SOUND,  AND  LIGHT 


FIG.  227 


one  vibrating  in,  and  one  normal  to,  the  plane  of  incidence.  Let 
the  intensity  of  each  of  these  components  be  represented  by  50 
(see  Fig.  227).  At  the  polarizing  angle  none  of  the  50  parts  which 
vibrate  in  the  plane  of  incidence  are  reflected,  while  photometric 

measurements  show  that 
about  16  per  cent  of  the 
light  which  is  vibrating 
perpendicular  to  the  plane 
of  incidence  is  reflected; 
that  is,  8  per  cent  of  the 
incident  beam  is  reflected 
at  the  polarizing  angle. 
Hence,  after  the  first  refraction,  the  transmitted  light  consists 
of  50  parts  vibrating  in  the  plane  of  incidence  and  42  parts 
vibrating  in  the  plane  perpendicular  to  the  plane  of  incidence. 
After  the  second  refraction  these  numbers  have  become  50  and 
35.3;  after  the  third  refraction,  50  and  29.7;  after  the  fourth, 
50  and  25,  and  so  on-.  After  passage  through  twelve  or  thirteen 
plates  the  transmitted  light  has  become  nearly 
plane  polarized  by  this  process,  the  plane  of 
its  vibrations  obviously  being  at  right  angles 
to  the  plane  of  vibration  of  the  reflected  light. 
A  pile  of  plates  of  this  sort  furnishes  a  very 
inexpensive  means  of  obtaining  plane  polar- 
ized light,  but  it  suffers  from  the  disadvantage 
that  the  polarization  is  not  quite  complete.  If 
no  light  whatever  were  absorbed  or  scattered 
by  the  glass,  the  transmitted  ray  would  be- 
come more  and  more  nearly  plane  polarized 
the  larger  the  number  of  plates,  but  in  prac- 
tice there  is  found  to  be  no  advantage  in 
increasing  the  number  of  plates  beyond  thir- 
teen or  fourteen. 

Experiment  4.   Replace  the  upper  mirror  mf  by  a  FIG.  228 

Nicol  prism  (Fig.  228),  the  construction  of  which 

will  be  considered  later,  and  looking  down  through  this  Nicol  at  the  image 
of  f  reflected  in  m,  rotate  the  Nicol  about  a  vertical  axis.  You  will  find 


POLAEIZED  LIGHT  321 

that  the  ray  mm'  is  cut  off  completely  by  the  Nicol  when  the  latter  is  in  a 
certain  position,  but  that  the  light  from  the  flame  is  transmitted  with 
maximum  brightness  when  the  Xicol  has  been  rotated  through  an  angle 
of  90°  from  this  position.  From  a  knowledge  of  the  plane  of  vibration 
of  the  ray  mm'  (Fig.  223)  decide  what  must  be  the  plane  of  vibration  of  a 
ray  with  respect  to  the  face  abed  of  the  Nicol,  in  order  that  it  may  be 
wholly  transmitted  by  the  latter,  and  mark  the  direction  of  this  transmit- 
ting plane  of  the  Xicol  by  an  arrow  drawn  on  the  face  of  mounting  contain- 
ing the  Xicol.  Henceforth  you  can  use  the  Xicol  as  a  detector  of  the  plane 
of  vibration  of  any  polarized  light  which  you  may  observe. 

198.  Polarization  by  double  refraction.  The  phenomena  which 
will  be  presented  in  the  following  experiments  were  discovered  in 
1670  by  the  Danish  physicist  Erasmus  Bartholinus  (1625—1698), 
who  first  noticed  the  fact  of  double  refraction  in  Iceland  spar, 
and  by  Huygens  (1629-1695)  in  1690,  who  first  noticed  the 
polarization  of  the  doubly  refracted  beams  produced  by  the  Ice- 
land spar,  and  first  offered  an  explanation,  of  double  refraction 
from  the  standpoint  of  the  wave  theory. 

Experiment  5.  Make  a  pinhole  in  a  piece  of  black  cardboard,  and  lay 
the  cardboard  on  a  piece  of  plane  glass  on  the  frame  h  (Fig.  223).  Some 
inches  beneath  this,  for  example  on  the  plate 
m  turned  into  the  horizontal  position,  lay  a 
piece  of  white  paper  and  illuminate  it  well. 
Then  lay  a  crystal  of  Iceland  spar  (Fig.  229) 
over  the  hole  in  the  cardboard.  Remove  m' 
and  look  vertically  down  upon  the  crystal. 
You  will  see  two  pin  holes  instead  of  one. 
Rotate  the  crystal  about  a  vertical  axis. 
One  image  will  remain  stationary,  while 
the  other  will  rotate  about  it.  FIG.  229 

That  the  image  which  remains  stationary  is  produced  by  light 
which  has  followed  the  usual  laws  of  refraction  is  evident  from 
the  fact  that  it  behaves  in  all  respects  as  it  would  if  viewed 
through  a  glass  plate.  The  image  which  rotates,  however,  must 
be  produced  by  light  which  has  followed  some  extraordinary  law 
of  refraction ;  for  although  it  has  passed  into  the  crystal  in  a 
direction  normal  to  the  bottom  face,  and  out  of  it  in  a  direction 
normal  to  the  top  face,  it  must  have  suffered  bending  inside  the 
crystal,  since  it  emerges  from  the  crystal  at  a  point  different  from 


322 


ELECTRICITY,  SOUND,  AND  LIGHT 


that  at  which  the  other  ray  emerges.  We  must  conclude,  then, 
that  a  ray  of  light  which  is  incident  upon  the  lower  face  of  such 
a  crystal  of  Iceland  spar  is  split  into  two  rays  by  the  spar,  and 
that  these  two  rays  travel  in  different  directions  through  the 
crystal.  The  ray  which  follows  the  ordinary  law  of  refraction  is 
called  the  ordinary  ray,  the  other  the  extraordinary  ray. 

Experiment  6.  To  find  the  direction  in  which  the  extraordinary  ray 
travels,  rotate  the  crystal  about  a  vertical  axis  above  the  pin  hole  and  note 
that  the  extraordinary  image  always  lies  in  the  line  connecting  the  ordi- 
nary image  and  the  solid  ob- 
tuse angle  of  the  face  which 
is  being  viewed,  and,  further, 
that  the  extraordinary  image 
is  always  on  that  side  of  the 
ordinary  which  is  away  from 
this  solid  obtuse  angle. 

It  will  be  evident,  then, 
from  these  experiments 
that  if  Figure  230  rep- 
resents a  section  of  the 
crystal  made  by  passing 
a  plane  normal  to  the 
top  and  bottom  faces  and 
through  the  two  solid  ob- 
tuse angles  of  the  rhomb, 
the  line  io  will  represent 

the  path  of  the  ordinary  ray  through  the  rhomb,  while  the  broken 
line  imne  will  represent  the  path  of  the  extraordinary  ray. 

Experiment  7.  In  order  to  determine  whether  the  ordinary  or  the  extraor- 
dinary ray  travels  the  faster  through  the  rhomb,  observe  again  the  two 
pin  holes,  or,  better,  observe  at  close  range  a  dot  on  a  piece  of  white  paper 
upon  which  the  crystal  lies,  and  note  which  image,  the  ordinary  or  the 
extraordinary,  appears  to  be  the  nearer  to  the  upper  face. 

This  will  evidently  correspond  to  the  ray  which  has  suffered 
the  largest  change  of  velocity  in  emerging  into  the  air  (see  p.  272); 
that  is,  it  will  correspond  to  the  ray  which  travels  more  slowly  in 
the  crystal.  This  will  be  found  to  be  the  ordinary  ray. 


FIG.  230 


POLARIZED  LIGHT  323 

Experiment  8.  If  you  can  obtain  a  crystal  which  has  been  cut  so  that 
its  top  and  bottom  faces  are  planes  which  are  at  right  angles  to  the  line 
ab  (Fig.  229)  which  connects  the  two  obtuse  angles  of  a  perfect  rhomb, 
that  is,  a  rhombohedroii  having  all  of  its  faces  equal,  view  the  pin  hole 
normally  through  this  crystal.  You  will  observe  that  there  is  now  but  one 
ray,  and  that  this  ray  does  not  change  position  upon  rotation  ;  that  is,  that 
it  behaves  in  the  ordinary  way. 

The  direction  of  the  line  connecting  the  two  obtuse  solid  angles 
of  a  crystal  all  of  whose  sides  are  equal  is  the  optic  axis  of  the 
crystal.  This  axis  is  not  a  line,  but  rather  a  direction.  Any  ray 
of  light  which  passes  through  the  crystal  in  a  direction  parallel  to 
the  line  ab  (Fig.  229),  that  is,  parallel  to  the  optic  axis,  does  not 
suffer  double  refraction. 

Experiment  9.  Place  the  Iceland  spar  again  over  the  pin  hole  in  the 
manner  indicated  in  Experiment  5,  and  view  the  two  images  through  the 
Nicol  prism  as  the  latter  is  rotated  about  a  vertical  axis.  You  will  find 
that  both  the  ordinary  and  the  extraordinary  images  consist  of  plane 
polarized  light,  but  that  the  planes  of  vibration  of  the  waves  which  pro- 
duce the  two  images  are  at  right  angles  to  one  another. 

Hence  we  may  conclude  that  the  Iceland  spar  has  in  some  way 
separated  the  incident  light  into  two  sets  of  vibrations,  one  of  which 
consists  of  all  the  components  of  the  initial  vibrations  which  were 
parallel  to  a  particular  plane  in  the  crystal,  while  the  other  con- 
sists of  all  of  the  components  of  the  initial  vibrations  which  were 
perpendicular  to  this  plane. 

Experiment  10.  With  the  aid  of  the  Nicol,  the  transmitting  plane  of 
which  you  determined  in  Experiment  4,  find  whether  the  ordinary  or  the 
extraordinary  ray  consists  of  vibrations  which  are  parallel  to  the  plane 
which  includes  the  optic  axis  of  the  crystal. 

You  will  find  that  it  is  the  extraordinary  ray  the  vibrations  of 
which  are  in  this  plane,  while  the  vibrations  of  the  ordinary  ray 
are  perpendicular  to  this  plane  (see  Fig.  230). 

199.  Theory  of  double  refraction.  The  elementary  theory  of 
double  refraction  is  as  follows : 

For  the  sake  of  simplicity  we  shall  confine  attention  to  the 
wave  form  in  a  single  plane  in  the  crystal,  namely  the  plane 
which  is  perpendicular  to  the  upper  and  lower  faces  of  the  crystal 


324 


ELECTRICITY,  SOUND,  AND  LIGHT 


FIG.  231 


and  includes  the  optic  axis.  This  is  called  the  principal  plane. 
It  is  the  plane  of  the  paper  in  Figures  231  and  232.  As  we  have 
already  seen,  any  incident  beam  of  light  which  passes  normally  into 

the  crystal  through  the 
hole  in  the  cardboard 
(see  Figs.  231  and  232) 
may  be  thought  of  as 
consisting  of  equal  vi- 
brations in  two  planes, 
one  perpendicular  to 
the  plane  of  the  paper 
and  the  other  parallel 
to  this  plane.  Let  us 
consider  these  two  vi- 
brations as  separated, 
so  that  we  may  treat 
of  one  in  Figure  231 
and  the  other  in  Fig- 
ure 232.  Let  us  suppose,  further,  that  any  vibrations  which  are 
parallel  to  the  direction  of  the  optic  axis  ab  pass  through  the 
crystal  with  greater 
facility,  that  is,  with 
greater  speed,  than  do 
vibrations  which  are 
perpendicular  to  this 
direction.  The  compo- 
nent in  the  plane  of  the 
paper  (see  Fig.  231)  of 
the  incident  vibrations 
will  give  rise  at  the 
boundary  mn  of  the 
crystal  to  transverse 
disturbances  which 
will  travel  outward 
in  all  directions  through  the  crystal.  The  portion  of  the  wave 
front,  however,  which  travels  at  right  angles  to  the  axis  ab,  that  is, 
in  the  direction  md  (Fig.  231),  will  have  its  vibrations  parallel  to 


FIG.  232 


POLAEIZED   LIGHT  325 

the  optic  axis,  while  the  portion  of  the  wave  front  which  travels 
in  the  direction  mg  will  have  its  vibrations  perpendicular  to  this 
axis.  If,  then,  vibrations  parallel  to  ab  travel  faster  than  do  such 
as  are  perpendicular  to  ab,  the  wave  which  originates  at  any  point 
on  mn  will  travel  faster  in  the  direction  md  than  in  the  direction 
mg,  and  will  consequently  have  an  elliptical  rather  than  a  spher- 
ical form,  the  longer  axis  of  the  ellipse  being  in  the  direction  at 
right  angles  to  the  optic  axis  ab.  The  envelope  of  all  the  ellipses 
which  originate  in  the  points  on  mn  will  be  the  line  m'nf:  The  beam 
will  therefore  travel  through  the  crystal  in  a  direction  other  than 
that  of  the  normal  to  its  wave  front ;  that  is,  in  the  direction  mm'. 
For  the  reason  given  in  section  154  (p.  242)  there  wrill  be  destruc- 
tive interference  at  all  points  outside  of  the  parallels  mm',  nnf. 

On  the  other  hand,  the  weaves  which  start  out  from  each  point 
on  mn  because  of  the  propagation  into  the  crystal  of  the  vibrations 
which  were  perpendicular  to  the  plane  of  the  paper  (see  Fig.  232) 
will  be  everywhere  perpendicular  to  the  optic  axis,  and  hence  will 
travel  with  equal  speeds  in  all  directions.  The  beam  will  there- 
fore follow  the  usual  law  of  refraction  and  will  travel  in  a  direc- 
tion at  right  angles  to  its  wave  front,  the  waves  from  each  point 
being  now  spheres  instead  of  ellipses. 

200.  Construction  of  the  Nicol  prism.  In  order  that 
the  light  which  is  transmitted  by  a  crystal  of  Iceland 
spar  may  consist  of  vibrations  in  one  plane  only,  it  is 
necessary  to  dispose  in  some  way  either  of  the  ordinary 
or  extraordinary  beam  so  as  to  prevent  it  from  passing 
through  the  crystal.  This  was  first  accomplished  in 
1828  by  the  German  physicist  Nicol  in  the  following 
way.  If  the  beam  be  (Fig.  233)  is  made  to  enter  the 
face  of  the  crystal  at  a  certain  oblique  angle,  the  ordi- 
nary ray,  being  refracted  more  than  the  extraordinary 
(see  Exp.  7),  will  travel  in  the  crystal  in  the  direction 
co,  for  example,  while  the  extraordinary  ray  will  take 
the  direction  ce.  Now  Nicol  cut  the  crystal  into  two 
parts  along  the  plane  aa,  and  then  cemented  the  parts  together 
again  with  Canada  balsam.  This  balsam  has  an  index  of  refrac- 
tion which  is  smaller  than  that  of  the  ordinary  ray,  but  larger 


326  ELECTRICITY,  SOUND,  AND   LIGHT 

than  that  of  the  extraordinary  ray ;  hence  it  was  possible,  by 
using  a  long  crystal  like  that  shown  in  the  figure,  to  choose  the 
plane  aa  so  that  the  ordinary  ray  would  be  totally  reflected  and 
absorbed  in  the  blackened  walls  of  the  crystal,  while  the  extraordi- 
nary ray  would  pass  through.  Several  modifications  of  this  device 
are  now  in  use,  but  they  are  all  essentially  the  same  in  principle. 
201.  Effects  produced  by  the  passage  of  polarized  light  through 
thin  crystals. 

Experiment  11.  Arrange  the  polarizing  apparatus  precisely  as  in  Fig- 
ure 223,  save  that  the  Nicol  prism  replaces  the  mirror  m'.  Rotate  the 
Nicol  until  the  flame  is  completely  extinguished.  Then  obtain  from  the 
instructor  a  so-called  half -wave  plate  (for  sodium  light)  of  mica  or  selenite 
and  place  it  on  the  slide  holder  Ji.  You  will  find  that,  in  general,  the  inser- 
tion of  the  mica  causes  the  light  to  reappear.  Rotate  the  mica  about  a 
vertical  axis  and  note  that  in  one  revolution  there  are  four  positions,  just 
00°  apart,  at  which  there  is  extinction.  These  are  the  positions  in  which 
the  plane  of  vibration  of  the  light  which  is  incident  upon  the  mica  is  either 
parallel  to  or  perpendicular  to  the  plane  containing  the  optic  axis  of  the 
mica.  Rotate  the  mica  in  a  horizontal  plane  until  it  is  just  45°  from  one 
of  these  positions  of  extinction.  Then  rotate  the  Nicol.  The  image  of  the 
flame  will  be  found  to  disappear  when  the  Nicol  has  been  rotated  through  90°. 

We  may  explain  this  phenomenon  as  follows.  In  Figure  234 
let  a  represent  the  plane  of  vibration  of  the  light  which  is  inci- 
dent upon  the  sheet  of  mica.  Let  l>  represent  the  two  vibrations 

a     ^    into  which  the  incident  vibration  is  decomposed  as 

soon  as  it  enters  the  crystal,  the  one,  e,  in  the  plane  of 
e  the  optic  axis,  and  the  other,  0,  perpendicular  to  this 
plane.  Since  one  of  these  two  waves  travels  faster 
through  the  mica  than  does  the  other,  a  sheet  may  be 
chosen  of  just  such  thickness  that  the  two  beams  will 
emerge  from  the  crystal  with  o  just  one-half  wave 
length  behind  e.  Since,  moreover,  the  beams  are  broad 
and  the  mica  very  thin,  o  and  e  are  not  separated  from 
one  another  as  they  were  in  the  case  of  the  two  nar- 

IT  IP     9*34. 

row  beams  used  in  Experiment  5.  The  same  portion  of 
the  ether  must  therefore  transmit  simultaneously  the  two  beams 
after  emergence  from  the  mica.  The  fact  that  one  of  them  has  lost 
one-half  wave  length  with  respect  to  the  other  has  been  indicated 


X. 
X 

t 


POLARIZED   LIGHT  327 

at  c  (Fig.  234),  by  changing  the  position  of  the  arrowhead  on  the 

line  which  represents  this  vibration  from  one  end  to  the  other. 

These  two  vibrations  will  recombine,  upon  emergence  into  air,  into 

one  single  vibration,  which  is  represented  by  Figure  234,  c?.    The 

light   which   emerges   from   the   crystal   will   therefore   be   plane 

polarized,  but  the  direction  of  its  vibration  will  be  at  right  angles 

to  the  direction  of  vibration   of  the   beam  when   it 

entered  the  crystal.    Since  the  Nicol  was  originally  set 

so  as  to  cut  out  vibrations  in  the  direction  shown  in  Q 

Figure  234,  a,  it  will,  of  course,  transmit  with  maxi-     ^      /^ 

mum  intensity  vibrations  in  the  direction  shown  in 

Figure  234,c£    In  order  to  extinguish  this  vibration  \7 

it  should  be  necessary  to  rotate  the  Nicol  through 

90°,  as  we  found  in  our  experiment  was  the  case. 

If  the  mica  plate  had  been  just  one  half  as  thick 
as  it  was,  the  two  components  would  have  emerged 
from  the  crystal  one-fourth  wave  length  instead  of 
one-half  wave  length  apart.  They  would  have  then  recombined 
into  a  circular  vibration  (see  Fig.  235),  since  two  equal  simple 
harmonic  forces  90°  apart,  acting  simultaneously  upon  the  same 
particle,  must  cause  it  to  describe  a  circular  path.  The  analyzing 
Nicol  should  obviously  transmit  the  same  fraction  of  this  circular 
vibration,  no  matter  into  what  plane  it  is  turned. 

Experiment  12.  Replace  the  half -wave  plate  by  one  half  as  thick,  that 
is,  by  a  quarter-wave  plate.  Set  it  at  first  so  that  it  is  45°  from  the  point  of 
extinction,  the  Xicol  being  set  in  the  position  corresponding-  to  extinction 
when  no  plate  is  interposed  ;  then  rotate  the  Xicol  and  note  that  no  change 
takes  place  in  the  intensity  of  the  transmitted  light  because  of  this  rotation. 

Light  of  this  sort  produced  by  passing  plane  polarized  light 
through  a  quarter-wave  plate  is  said  to  be  circularly  polarized. 

If  the  plate  is  of  such  thickness  as  to  produce  a  retardation  (of 
one  component  with  respect  to  the  other)  of  one  wave  length,  two 
wave  lengths,  three  wave  lengths,  and  so  on,  the  light  will  obvi- 
ously emerge  from  the  crystal  as  plane  polarized  light  vibrating  in 
the  same  plane  as  that  in  which  it  entered  the  crystal.  Hence  it 
will  be  completely  cut  out  by  the  Nicol  when  the  latter  is  set  in  the 
position  of  extinction  for  the  case  in  which  no  crystal  is  interposed. 


328 


ELECTRICITY,  SOUND,  AND  LIGHT 


x; 

X 

0 


FIG.  236 


If  the  plate  is  of  such  thickness  as  to  cause  retardation  inter- 
mediate between  a  quarter  wave  and  a  half  wave,  or  a  half  wave 
and  a  whole  wave,  or  a  whole  wave  and  a  wave  and  a 
quarter,  and  so  on,  the  light  which  emerges  from  the 
crystal  is  elliptically  polarized,  that  is,  the  vibrations 
of  the.  ether  particles  take  place  in  the  form  of  an 
ellipse  (see  Fig.  236).  The  major  and  minor  axes 
of  this  ellipse  may  be  easily  found  by  observing  in 
what  direction  the  analyzing  Nicol  must  be  turned 
in  order  to  obtain  a  maximum  or  a  minimum  of 
transmitted  light. 

202.  Colors  produced  by  thin  crystals  in  polar- 
ized light. 

Experiment  13.  Set  the  polariscope  in  a  window  in  the  position  shown 
in  Figure  237,  the  black  paper  being  removed  from  the  mirror  n.  If  m  has 
precisely  the  same  inclination  which  "was  given  it  in  Experiment  1  (Fig.  223) , 
then,  when  the  polariscope  is  so  turned  that  the 
prolongation  of  the  line  ma  meets  the  clear  sky, 
the  white  light  from  the  sky  will  strike  the  lower 
side  of  m  at  the  polarizing  angle,  be  reflected  to 
the  mercury  mirror  w,  and  return  with  little  loss 
as  a  plane  polarized  beam  to  the  Nicol  N.  Set  N 
so  that  this  beam  is  extinguished.  Place  a  sheet 
of  mica  about  twice  as  thick  as  a  half-wave  plate 
for  sodium  light  upon  h  and  turn  it  until  it  is  just 
45°  from  a  position  of  extinction.  When  viewed 
through  the  Nicol  it  will  be  found  to  be  brilliantly 
colored.  Rotate  the  Nicol  slowly  and  notice  that 
a  rotation  of  45°  causes  the  color  to  disappear, 
but  that  a  rotation  of  90°  causes  a  color  which 
is  the  complement  of  the  first  color  to  appear. 
Further  rotation  through  90°  will  cause  the  first 
color  to  return,  and  so  on.  When  sheets  of  mica 
of  different  thicknesses  are  used  different  colors 
will  be  produced,  but  a  rotation  of  the  Nicol 
through  90°  will  always  cause  the  color  to  change 
to  that  of  the  complement  of  the  original  color.  -^  007 

In  order  to  understand  the  cause  of  this  phenomenon  suppose, 
for  simplicity,  that  the  mica  is  just  thick  enough  to  produce  a 
retardation  of  one-half  wave  length  of  the  longest  red  wave. 


POLARIZED  LIGHT  329 

Since  the  shortest  violet  waves  have  about  one  half  the  wave 
length  of  the  longest  red,  this  same  plate  will  produce  a  retarda- 
tion in  the  violet  of  one  whole  wave  length.  The  violet  wave  will 
therefore  emerge  from  the  crystal  with  both  of  its  components  in 
the  same  phase,  and  these  components  will  recombine  into  a  plane 
vibration  precisely  like  that  which  entered  the  crystal,  namely 
the  vibration  represented  in  Figure  234,  a.  The  red  ray,  however, 
will  emerge  from  the  crystal  with  one  of  its  components  one-half 
wave  length  behind  the  other,  and  these  two  components  will 
recombine  into  a  vibration  at  right  angles  to  that  of  the  entering 
ray,  namely  one  of  the  form  shown  in  Figure  234,  d.  If,  then, 
the  Nicol  is  in  the  position  for  extinction  when  no  crystal  is 
interposed,  it  will  cut  out  all  of  the  violet  in  the  incident  white 
light  and  transmit  all  of  the  red,  so  that  if  these  red  and  violet 
waves  fell  alone  upon  the  crystal,  a  rotation  of  the  Nicol  would 
cause  red  and  violet  to  appear  alternately.  As  a  matter  of  fact, 
however,  if  the  incident  light  is  white,  all  of  the  colors  between 
the  red  and  the  violet  will  be  present,  and  the  vibrations  of  the 
transmitted  light  which  correspond  to  them  will  be  ellipses  of 
some  form.  However,  the  wave  lengths  which  are  close  to  the 
red,  namely  orange  and  yellow,  will  be  largely  transmitted  by  the 
Nicol  along  with  the  red,  and  will  have  but  small  components  to 
be  transmitted  with  the  violet,  while  the  wrave  lengths  near  the 
violet,  namely  the  blues  and  the  shorter  greens,  will  be  mainly 
transmitted  with  the  violet.  Hence  the  light  which  passes  through 
the  Nicol  in  its  first  position  will  be  some  shade  of  red,  because  it 
will  have  most  of  the  shorter  wave  lengths  subtracted  from  it ; 
while,  when  the  Nicol  is  turned  through  90°,  all  of  the  wave 
lengths  which  were  before  cut  out  will  be  transmitted.  The  color 
will  therefore  be  exactly  the  complement  of  the  first  color,  that  is, 
it  will  be  some  shade  of  blue. 

A  crystal  which  is  too  thin  to  produce  one-half  wave  length 
retardation  of  the  shortest  visible  rays,  namely  the  violet,  cannot 
show  any  marked  color  effects  in  polarized  light,  since  no  wave 
length  can  be  entirely  cut  out  for  any  position  of  the  Nicol.  On 
the  other  hand,  a  crystal  which  is  so  thick  as  to  produce  retarda- 
tion of  very  many  wave  lengths  of  any  one  color  will  produce  also 


330  ELECTRICITY,  SOUND,  AND  LIGHT 

a  retardation  of  an  exact  number  of  wave  lengths  for  each  of  many 
other  colors  scattered  throughout  the  spectrum.  These  colors  will 
all  be  cut  out  by  the  Nicol,  and  the  transmitted  light  will  like- 
wise consist  of  wave  lengths  which  are  taken  from  all  parts  of 
the  spectrum,  and  will  therefore  reproduce  the  effect  of  white 
light.  Hence  these  color  phenomena  in  polarized  light  can  be 
observed  only  with  crystals  which  produce  a  small  number  of 
ivave  lengths  of  retardation.  By  scraping  crystals  down  to  proper 
thicknesses  in  different  parts,  color  patterns  of  much  beauty  are 
often  produced  when  the  crystals  so  treated  are  viewed  in  the 
polarized  light.  All  the  colors,  of  course,  change  to  the  comple- 
ments upon  rotation  of  the  Nicol  through  90°. 

Experiment  14.  Place  a  iiiimber  of  these  designs  in  selenite  or  mica 
upon  the  slide  holder  h,  and  observe  the  appearance  of  the  complementary 
colors  in  different  portions  of  the  design  as  the  Nicol  is  rotated. 

Experiment  15.  Observe  in  convergent  polarized  light  a  crystal  of  Iceland 
spar,  say  1  mm.  thick,  the  upper  and  lower  faces  of  which  are  planes  per- 
pendicular to  the  optic  axes.  The  beam  of  convergent  light  is  most  easily 

obtained  by  placing  the  crystal 
very  close  to  the  Nicol  in  the  ar- 
rangement of  Figure  237,  so  that 
the  observer  looks  down  through 
the  crystal  upon  a  field  of  con- 
siderable width,  from  all  parts 
of  which  polarized  light  is  ap- 

F,o.  238  F.o.  239  P™cl>ing  the  eye     If  the  Nicol 

was  originally  set  tor  extinction, 

you  will  observe  a  dark  center  surrounded  by  a  series  of  brilliantly  colored 
rings  upon  which  is  superposed  a  black  cross  (see  Fig.  238).  Rotating  the 
Nicol  will  cause  the  black  cross  to  change  to  a  white  one,  and  all  of  the 
colors  to  change  to  their  complements  (see  Fig.  239). 

These  effects  may  be  explained  as  follows.  The  central  rays 
pass  through  the  crystal  in  a  direction  parallel  to  the  optic  axis. 
They  therefore  suffer  no  resolution  into  ordinary  and  extraordi- 
nary components,  and  hence  no  change  in  the  character  of  their 
vibration.  They  are  cut  out  by  the  analyzing  Nicol,  hence  the 
black  center.  The  rays,  however,  which  converge  upon  the  eye 
after  passing  through  the  outer  edges  of  the  crystal  have  trav- 
eled in  directions  slightly  oblique  to  the  axis,  and  have  therefore 


POLARIZED  LIGHT  331 

suffered  decomposition  into  ordinary  and  extraordinary  rays,  which 
have  undergone  different  retardations.  A  given  retardation  of  one 
ray  with  respect  to  the  other  corresponds  to  a  given  color  pre- 
cisely as  explained  above.  A  given  color  must,  of  course,  be  sym- 
metrically distributed  about  the  axis  of  the  converging  beam,  since 
the  thickness  of  the  crystal  is  so  distributed ;  hence  the  concentric 
rings  of  color.  The  black  cross  is  superposed  upon  these  rings  be- 
cause in  two  particular  planes,  namely  those  for  which  the  inci- 
dent vibration  is  respectively  in  and  perpendicular  to  the  plane 
containing  the  axis  and  the  ray,  even  these  oblique  rays  are  not 
split  up  into  components,  but  pass  through  vibrating  in  their 
original  direction,  and  are  therefore  cut  out  by  the  Mcol.  Upon 
rotating  the  Nicol  through  90°  all  of  these  extinguished  rays  are, 
of  course,  transmitted  ;  hence  the  white  cross. 
203.  Rotary  polarization. 

Experiment  16.  Arrange  the  polarizing  apparatus  as  in  Figure  223,  save 
that  mf  is  replaced  by  a  Xicol,  and  place  upon  h  a  crystal  of  quartz,  say 
5  mm.  thick,  the  upper  and  lower  faces  of  which  are  made  by  planes  which 
are  at  right  angles  to  the  optic  axis  of  the  quartz.  When  the  Xicol  is  set 
for  extinction  the  introduction  of  the  quartz  into  the  path  of  the  beam 
will  be  found,  in  general,  to  cause  the  extinguished  image  of  the  flame  to 
reappear.  Rotate  the  Xicol,  and  measure  the  amount  of  rotation  required 
to  cause  the  yellow  flame  to  disappear  again.  According  to  accepted  results 
this  rotation  for  sodium  light  should  be  21.7°  per  millimeter  of  thickness 
of  the  quartz.  Replace  the  sodium  flame  by  the  ordinary  violet  flame  of 
the  Bunsen  burner  and  repeat,  rotating  this  time  until  all  trace  of  the  violet 
color  in  the  flame  has  disappeared.  The  rotation  will  be  found  to  be  nearly 
double  that  found  for  sodium  light.  The  rotation  in  the  case  of  light  filtered 
through  red  glass  will  be  found  to  be  less  than  that  in  the  case  of  yellow  light. 

These  experiments  show,  first,  that  plane  polarized  light  which 
passes  through  quartz  in  the  direction  of  its  optic  axis  remains 
plane  polarized  after  transmission,  and,  second,  that  the  plane  of 
polarization  of  the  light  is  rotated  by  the  quartz,  the  amount  of 
the  rotation  being  greater  for  the  short  wave  lengths  than  for  the 
long.  The  discovery  that  quartz  is  able  to  produce  these  effects 
was  made  by  Arago  in  1811. 

From  the  difference  in  the  amount  of  rotation  of  different  colors 
it  follows  that  if  plane  polarized  white  light  is  incident  upon  the 


332  ELECTRICITY,  SOUND,  AND  LIGHT 

lower  face  of  the  crystal  of  quartz,  the  light  which  is  transmitted 
by  the  analyzing  Nicol  will  be  colored,  since  this  Nicol  will  extin- 
guish completely  only  those  vibrations  which  are  perpendicular  to 
its  transmitting  plane.  It  follows,  further,  that  if  the  Nicol  is 
rotated  through  90°,  all  of  the  components  of  the  white  light 
which  were  before  extinguished  will  be  now  transmitted  and  vice 
versa,  and  hence  that  the  rotation  through  90°  will  cause  the 
color  to  change  to  the  complement  of  the  first  color. 

Experiment  17.  Verify  the  above  predictions  by  setting  the  polariscope 
so  that  light  from  the  sky  falls  upon  it  in  the  manner  indicated  in  Fig- 
ure 237.  To  show  that  the  lights  transmitted  by  the  analyzer  in  planes 
90°  apart  are  complementary,  it  is  best  to  replace  the  Nicol  by  a  thick 
crystal  of  Iceland  spar,  or  some  other  form  of  double-image  prism,  so  that 
both  of  the  lights  to  be  compared  may  be  transmitted  at  once.  As  the 
analyzer  is  rotated,  the  overlapping  portions  of  the  images  will  be  found 
to  maintain  the  color  of  white  light,  while  the  opposite  non-overlapping 
portions  will  be  of  complementary  colors. 

It  has  been  found  that  there  are  two  kinds  of  quartz  crystals, 
one  of  which  produces  rotation  to  the  right,  the  other  to  the  left. 
This  difference  in  optical  behavior  corresponds  also  to  a  difference 
in  crystalline  structure  which  makes  it  easy  to  distinguish  the 
so-called  right-handed  from  the  left-handed  quartz  crystals  with- 
out actually  making  the  optical  test. 

Furthermore,  it  was  discovered  by  Biot,  in  1815,  that  there 
are  certain  liquids  which  possess  the  same  property  shown  in  the 
above  experiments  by  quartz.  Of  these,  solutions  of  cane  sugar 
have  received  most  attention,  for  the  reason  that  the  amount  of 
rotation  produced  by  a  column  of  sugar  solution  of  fixed  length  is 
taken  as  the  commercial  test  of  the  strength  of  the  solution.  As 
in  the  case  of  quartz,  there  are  found  to  be  two  kinds  of  cane 
sugar  of  precisely  the  same  chemical  constitution,  but  of  slightly 
different  crystalline  form,  which  rotate  the  plane  of  polarization  in 
opposite  directions.  The  form  which  rotates  to  the  right  is  called 
dextrose,  the  other  levulose.  It  is  possible  to  convert  dextrose  to 
levulose  sugar  by  acting  upon  it  with  hydrochloric  acid.  This 
conversion  is  actually  made  in  sugar  testing,  the  rotation  due  to 
the  conversion  being  the  quantity  directly  measured. 


CHAPTEK  XXVIII 
RADIO-ACTIVITY 

204.  Types  of  radiation.   Up  to  about  1895  the  only  types  of 
radiation  which  were  generally  recognized  were  those  comprised 
under  the  two  heads, — sound  waves  and  ether  waves.    Ether  waves 
included  light  waves,  of  wave  length  .00038  mm.  to  .00076  mm.; 
ultraviolet  waves,  studied  by  means  of  their  photographic  effects 
and   having   wave   lengths   extending   from   .00038   down   to   an 
unknown  limit,  but  definitely  explored  down  to  .0001 ;  ultrared 
rays,  studied  by  means  of  their  heating  effects,  and  having  wave 
lengths  extending  from  .00078  mm.  up  to  an  unknown  limit,  but 
explored  up  to  .06  mm.;  and  electrical  waves  studied  by  means 
of  electrical  resonators,  and  having  observed  wave  lengths  extend- 
ing from  3  mm.  up  to  infinity.    All  of  these  forms  of  ether  waves 
have  been  definitely  proved  to  travel  with  the  same  velocity  in  a 
vacuum  and  to  be  essentially  alike  except  in  the  matter  of  wave 
length. 

Since  1895  three  new  types  of  radiation  have  gained  recognition. 
They  are  cathode  rays,  X  rays,  and  the  radiations  produced  by 
radio-active  substances.  Cathode  rays  were  observed  as  early  as 
1859  and  named  in  1876,  but  their  character  was  not  understood 
until  1897. 

205.  Cathode  rays.    When  an  electrical   discharge    is    forced 
through  a  glass  tube  from  which  the  air  is  beiog  exhausted,  the 
appearance  of  the  discharge  is  found  to  vary  continuously  as  the 
exhaustion   progresses.    When  the  pressure  within  the  tube  has 
been  reduced  to  say  .01  mm.  of  mercury,  a  greenish  fluorescence  is 
found  to  have  made  its  appearance  on  the  walls  of  the  tube  about 
the  cathode.     As  the  exhaustion  is  carried  still  farther  all  appear- 
ance of  a  discharge  through  the  gas  within  the  tube  vanishes  and 
the  walls  become  suffused  with  a  greenish  fluorescence. 


334 


ELECTKICITY,  SOUND,  AND  LIGHT 


This  fluorescence  seems  to  be  due  to  some  sort  of  radiation 
which  is  emitted  by  the  cathode  in  a  direction  normal  to  its 
surface ;  for  whenever  an  object  is  placed  within  the  tube  a  sharp 

shadow  of  it  is  found  to 
be  cast  upon  the  walls 
at  a  point  directly  op- 
posite the  cathode,  no 
matter  where  the  anode 
may  be  placed.  This 
is  well  illustrated  by 
an  experiment  devised 
by  Sir  William  Crookes 
in  1879,  and  repre- 
sented in  Figure  240, 
a  shadow  of  the  Mal- 
tese cross  M  being  evident  upon  the  wall  of  the  tube  opposite 
the  cathode  C. 

206.  Theory  of  cathode  rays.  The  nature  of  these  so-called 
cathode  rays  was  a  subject  of  much  dispute  between  the  years 
1880  and  1895.  Some  thought 
them  to  be  some  type  of  ether 
waves  emitted  by  the  cathode, 
while  others,  following  the  lead 
of  Crookes,  were  convinced,  that 
they  consisted  of  material  par- 
ticles projected  with  large  veloc- 
ities from  the  surface  of  the 
cathode.  The  most  convincing 
evidence  in  favor  of  the  latter 
view  was  found  in  the  fact, 
brought  out  clearly  by  Crookes 
in  1879,  that  the  cathode  rays 
are  deflected  by  a  magnet  in  FIG.  241 

precisely  the  way  in  which  neg- 
atively charged  particles  projected  normally  from  the  surface  of 
the  cathode  ought   to  be  deflected.    The  experiment  illustrating 
this  deflection  is  shown  in  Figure  241.    The  cathode  beam  from 


EADIO-ACTIVITY 


335 


c,  the  direction  of  which,  after  it  passes  through  the  slit  e,  is 
indicated  by  the  fluorescence  which  it  produces  in  the  zinc  sul- 
phide screen  ad,  along  which  it  grazes  as  it  progresses  through 
the  tube,  is  deflected  by  the  magnet  ns,  in  the  position  shown, 
precisely  as  the  motor  rule  indicates  that  it  should  be  deflected  if 
the  cathode  beam  consisted  of  a  negative  current  flowing  from  the 
bottom  toward  the  top  of  the  tube,  that  is,  of  a  positive  current 
flowing  from  top  to  bottom. 

The  strongest  objection  to  the  projected-particle  theory  was 
made  on  the  ground  of  experiments  conducted  by  Lenard,  in  Bonn, 
Germany,  in  1893.  These  experiments  show  that  the  cathode  rays 
can  pass  out  of  the  vacuum  tube  through  a  thin  aluminum  win- 
dow and  travel  several  centimeters  in  air  at  atmospheric  pressure 
before  being  absorbed.  Figure  242  shows  the  tube  commonly  used 
for  repeating  this 
experiment.  In 
this  figure  c  rep- 
resents the  cath- 
ode, A  the  anode, 
in  this  case  a  cyl- 
inder behind  the 
cathode,  w  a  very 

minute  and  very  thin  piece  of  aluminum  foil,  and  E  the  cathode 
rays  diffusing  in  air  at  atmospheric  pressure  in  the  manner  indi- 
cated. The  presence  of  these  rays  at  any  point  is  detected  by  the 
fluorescence  which  they  produce  in  an  exploring  zinc  sulphide  screen. 

This  experiment  shows  that  whatever  cathode  rays  may  be,  they 
are  able  to  pass  through  a  sheet  of  aluminum  which  wiU  not  per- 
mit the  passage  of  the  molecules  of  gas,  since  otherwise  the  vacuum 
in  the  tube  could  not  be  maintained ;  and  that  they  are  also  able 
to  pass  through  the  million  or  so  of  molecules  which  any  straight 
line  which  is  drawn  through  several  centimeters  of  air  at  atmos- 
pheric pressure  must  encounter.  It  is  evident  that  this  experiment 
is  not  easily  reconciled  with  the  projected-particle  theory  if  the 
projected  particles  are  assumed  to  be  atoms  or  molecules  of  ordi- 
nary matter ;  but  all  difficulty  vanishes  if  we  assume  that  the  pro- 
jected particles  are  exceedingly  small  in  comparison  with  ordinary 


FIG.  242 


336  ELECTRICITY,  SOUND,  AND  LIGHT 

atoms,  and  if  we  assume  further  that  the  atoms  of  ordinary  matter 
form  some  sort  of  infinitesimal  stellar  systems,  the  distances  be- 
tween whose  members  are  large  in  comparison  with  the  size  of  the 
members  themselves. 

In  1897  it  was  definitely  shown,  both  by  J.  J.  Thomson  and 
Lenard,  that  the  cathode  rays  are  deflected,  not  only  by  a  magnetic 
field,  but  also  by  an  electrostatic  field,  in  precisely  the  way  in 
which  they  should  be  deflected  if  they  consist  of  negatively  charged 
particles.  Furthermore,  by  comparing,  with  the  aid  of  mathemat- 
ical analysis,  the  amounts  of  deflection  produced  by  magnetic  and 
electric  fields  of  known  strengths  it  was  found  that  the  mass  of  the 
projected  particles  of  the  cathode  rays,  if  this  theory  were  adopted, 
came  out  but  1/1800  of  the  mass  of  the  smallest  known  atom, 
namely  that  of  hydrogen.  This  epoch-making  discovery  removed 
all  the  serious  difficulties  of  the  projected-particle  theory.  The 
ether  theory,  on  the  other  hand,  has  now  been  entirely  abandoned, 
since  it  does  not  seem  possible  to  reconcile  it  in  any  way  with  the 
magnetic  and  electrostatic  deflectibility  of  the  rays.  Furthermore, 
the  cathode  rays  were  definitely  proved  in  1896,  by  Perrin,  of  France, 
to  impart  negative  charges  to  any  objects  upon  which  they  fall. 
This  accords  with  the  projected-particle  hypothesis,  but  it  is  wholly 
unlike  any  property  possessed  by  known  forms  of  ether  waves. 

207.  Hypothesis  as  to  the  nature  of  the  atom.  The  acceptance 
of  the  projected-particle  theory  as  to  the  nature  of  cathode  rays 
makes  it  necessary  to  assume  that  these  minute  negatively  charged 
corpuscles,  or  electrons  as  they  have  been  named,  are  constituents 
of  the  atoms  of  all  kinds  of  matter,  since  cathode  rays  of  the  same 
magnetic  and  electric  deflectibility  can  be  produced  with  all  sorts 
of  electrodes  and  all  sorts  of  residual  gases  in  the  discharge  tube. 
This  assumption  met  with  very  striking  support  in  a  discovery 
announced  by  Zeeman,  of  Amsterdam,  in  1897.  This  observer 
found  that  the  spectral  lines  produced  by  incandescent  vapors 
become  doubled,  and  otherwise  modified,  when  the  incandescent 
vapor  which  is  being  examined  is  placed  between  the  poles  of  a 
powerful  electro-magnet.  From  the  character  and  magnitude  of 
this  effect  it  was  found  that  the  radiations  emitted  by  these  incan- 
descent vapors  must  be  produced  by  the  vibrations,  or  rotations 


RADIO-ACTIVITY  337 

about  a  center,  of  negatively  rather  than  positively  charged  par- 
ticles, and  that,  within  the  limits  of  experimental  error,  the  mass 
of  these  rotating  negative  particles  is  precisely  that  of  the  cathode- 
ray  particles. 

In  view,  then,  of  cathode-ray  phenomena,  the  Zeeman  effect,  and 
radio-activity  (sect.  210),  the  atoms  of  all  substances  are  now 
regarded  as  containing  negatively  charged  particles,  or  electrons,  of 
about  1/1800  the  mass  of  a  hydrogen  atom.  These  electrons, 
escaping  from  the  material  of  the  cathode  under  the  influence  of 
the  powerful  electric  fields  produced  in  cathode-ray  tubes,  stream 
away  in  straight  lines  from  the  surface  of  the  cathode  with  veloci- 
ties which  have  been  definitely  measured  and  found  to  lie  between 
one  tenth  and  one  third  the  velocity  of  light  (3  x  1010  cm.  per 
second),  the  velocity  being  determined  solely  by  the  strength  of 
the  field  which  is  employed  to  produce  the  discharge. 

In  order  to  account  for  the  fact  that  the  absorbing  power  of 
different  kinds  of  matter  for  the  cathode  rays  is  nearly  directly 
proportional  to  density,  it  is  sometimes  assumed  that  the  atoms 
of  all  substances  are  merely  aggregates  of  electrons,  so  that  the 
total  weight  of  a  cubic  centimeter  of  any  substance  is  determined 
solely  by  the  number  of  electrons  contained  in  it,  and  the  total  ab- 
sorbing power  of  this  cubic  centimeter  for  cathode  rays  is  deter- 
mined by  the  number  of  electrons  which  a  given  cathode-ray 
particle  would  encounter  in  passing  through  this  cubic  centimeter. 
This  view  would  be  in  accordance  with  Lenard's  experiment  on 
the  absorbing  powers  of  different  substances  for  cathode  rays,  but 
there  is  not  enough  evidence  to  substantiate  it  fully,  and  it  must 
be  held  at  present,  therefore,  merely  as  a  speculation. 

Since,  however,  the  atoms  certainly  contain  electrons,  and  since 
these  electrons  are  negatively  charged,  in  order  to  account  for  neu- 
tral atoms  we  must  assume  a  total  positive  charge  in  each  atom 
equal  to  the  sum  of  the  negative  charges  carried  by  its  electrons. 
Since  no  definite  evidence  has  as  yet  been  brought  forward  that 
positive  bodies  of  the  minute  mass  of  the  negative  electrons  are 
ever  set  free  from  atoms,  it  has  been  suggested  by  J.  J.  Thomson 
that  the  atom  may  consist  of  a  nucleus  of  positive  electricity  about 
which,  or  within  which,  the  negative  electrons  are  rotating. 


338  ELECTKICITY,  SOUND,  AND  LIGHT 

208.  The  ionization  of  gases.  One  of  the  most  notable  proper- 
ties possessed  by  cathode  rays  is  the  property  of  rendering  any  gas 
through  wliich  they  pass  a  conductor  of  electricity.  Thus  if  a 
charged  electroscope  is  placed  anywhere  within  a  foot  or  so  of  the 
window  w  (Fig.  242),  it  is  found  to  lose  its  charge  rapidly.  The 
way  in  wliich  this  discharge  takes  place  has  been  very  convinc- 
ingly shown  to  be  as  follows.  The  cathode  rays  in  passing  through 
the  gas  produce  in  it,  in  some  way,  both  positively  and  negatively 
charged  particles.  If,  then,  the  electroscope  is  positively  charged, 
for  example,  the  negative  particles  produced  in  the  gas  by  the 
cathode  rays  are  drawn  to  the  positively  charged  electroscope, 
give  up  their  charges  to  it  as  soon  as  they  touch  it,  and  thus  reduce 
its  positive  charge.  If  the  electroscope  is  negatively  charged,  it  is 
the  positive  particles  of  the  gas  which  discharge  it. 

The  origin  of  these  positive  and  negative  particles  is  supposed 
to  be  as  follows.  The  neutral  molecules  of  the  gas,  each  of  which 
contains  negative  electrons  and  an  amount  of  positive  electricity 
equal  to  the  sum  of  the  negative  charges  on  its  electrons,  have  one 
or  more  of  these  electrons  knocked  off  by  the  bombardment  of  the 
rapidly  moving  cathode-ray  particles.  The  loss  of  an  electron  by 
a  neutral  molecule  of  the  gas  leaves  it  positively  charged.  This 
positively  charged  molecule,  with  a  few  neutral  molecules  which  it 
probably  attaches  to  itself,  constitutes  one  of  the  positive  ions  of 
the  gas.  On  the  other  hand,  the  electrons,  which  have  been  knocked 
off  from  the  neutral  molecules,  probably  soon  attach  themselves  to 
other  neutral  molecules  and  thus  impart  negative  charges  to  them. 
These  negatively  charged  molecules,  with  neutral  molecules  which 
become  attached  to  them,  constitute  the  negative  ions  of  the  gas. 
When  two  of  these  oppositely  charged  ions  approach  near  enough 
to  each  other,  they  doubtless  are  drawn  together,  their  opposite 
charges  neutralized,  and  the  molecules  restored  to  their  original 
condition.  It  is  in  this  way  at  least  that  we  attempt  to  account  for 
the  fact  that  the  gas  rapidly  loses  its  conducting  power  when  the 
cathode  rays  are  removed.  If,  however,  the  ions  are  formed  in  a 
sufficiently  strong  electrical  field,  they  may  be  drawn  to  the  elec- 
trodes before  they  have  any  opportunity  to  unite  with  any  other 
oppositely  charged  ions.  In  this  case  the  rate  at  which  either  of 


BADIO-ACTIVITY  339 

the  electrodes  loses  its  charge  is  a  measure  of  the  total  number 
of  ions  formed  per  second. 

The  ionization  of  a  gas  then  differs  from  the  ionization  of  liquids, 
studied  in  Chapter  XVI,  in  the  following  respects.  While  the 
ionization  of  liquids  occurs  spontaneously,  the  ionization  of  gases 
occurs  only  through  the  action  of  an  outside  ionizing  agent.  The 
ionization  of  liquids  occurs  only  in  the  case  of  a  mixture  of  differ- 
ent substances,  that  is,  in  the  case  of  solutions,  but  the  ionization 
of  a  gas  occurs  as  well  with  simple,  homogeneous  gases  as  with 
mixed  ones. 

209.  X  rays.  It  was  in  December,  1895,  that  Professor  Eont- 
gen  announced  the  discovery  of  a  new  type  of  radiation  which  he 
named  X  rays.  These  rays  have  their  origin  in  cathode-ray  tubes, 
and  are  like  cathode  rays  in  the  fluorescent,  photographic,  and  ion- 
izing effects  which  they  produce,  but  differ  from  cathode  rays, 
first,  in  that  they  possess  a  very  much  higher  penetrating  power ; 
second,  in  that  they  are  not  deflected  in  the  slightest  degree  either 
by  a  magnetic  or  an  electrostatic  field ;  third,  in  that  they  do 
not  impart  negative  charges  to  bodies  upon  which  they  fall ;  and 
fourth,  in  that  different  bodies  do  not  absorb  them  in  amounts 
proportional  to  the  absorption  coefficients  of  these  same  bodies 
for  cathode  rays. 

That  the  X  rays  have  a  much  greater  penetrating  power  than  the 
cathode  rays  is  shown  by  the  fact  that  while  the  latter  can  only 
be  obtained  outside  a  cathode-ray  tube  by  passing  them  through 
a  very  thin  aluminum  window,  as  Lenard  did,  the  former  pass  with 
considerable  ease  through  the  glass  walls  of  the  ordinary  X-ray 
tube.  It  is  because  of  their  relatively  great  penetrating  power, 
and  because  of  the  fact  that,  in  general,  they  are  absorbed  in  dif- 
ferent amounts  by  substances  which  have  different  densities,  that 
it  is  possible  to  take  with  them  shadow  pictures  of  the  bones  of 
the  body,  or  of  any  objects  of  different  density  inclosed  in  cover- 
ings opaque  to  ordinary  light.  It  was  doubtless  on  account  of  this 
property  that  the  rays  attracted  so  immediate  and  so  widespread 
attention. 

Subsequent  study  of  X  rays  has  proved  conclusively  that  they 
originate  in  the  surface  of  any  body  upon  which  the  cathode  rays 


340  ELECTRICITY,  SOUND,  AND  LIGHT 

fall.  In  the  ordinary  X-ray  tube  a  platinum  plate  P  (Fig.  243)  is 
used  as  a  target  for  the  cathode  rays  which  are  emitted  at  right 
angles  to  the  surface  of  the  concave  cathode  C  and  come  to  a 
focus  at  a  point  on  this  plate.  This  is  the  point  at  which  the 
X  rays  are  generated  and  from  which  they  radiate  in  all  direc- 
tions. This  statement  can  be  readily  verified  by  turning  about 
an  X-ray  tube  in  action,  placed  at  a  distance  of  a  few  feet  from 
a  charged  electroscope,  and  noticing  that  the  electroscope  loses 
its  charge  rapidly  when  it  is  in  any  position  such  that  rays 

radiating  from  the  middle  of 
the  front  face  of  P  could  fall 
upon  it,  but  that  the  loss  of 
charge  ceases  as  soon  as  the 
tube  is  turned  into  such  a  po- 
sition that  rays  coming  from 

this  point  could  not  fall  upon 
FIG.  243  ^        i 

the  electroscope. 

The  nature  of  the  X  rays  is  not  yet  definitely  settled,  but  it  is 
very  generally  believed  that  they  are  very  violent  ether  pulses  set 
up  by  the  sudden  stoppage  of  the  cathode  rays  as  they  strike  upon 
the  target  P.  If  this  theory  is  correct,  we  must  regard  the  ioniza- 
tion  which  the  X  rays  produce  as  brought  about  by  the  shaking 
loose  of  electrons  from  the  atoms  of  air  because  of  the  suddenness 
of  the  ether  pulses  which  strike  them.  This  theory  is  in  accord 
with  the  fact  that  the  rays  show  no  regular  reflection,  refraction, 
or  polarization. 

210.  Discovery  of  radio-activity.  At  the  time  of  the  discovery 
of  X  rays,  and  before  their  origin  was  known,  it  was  surmised  that 
they  might  be  in  some  way  connected  with  the  fluorescence  of 
the  walls  of  the  cathode-ray  tube,  and  since  physicists  were  look- 
ing for  other  sources  of  the  rays,  it  occurred  to  Henri  Becquerel, 
of  Paris,  that  the  mineral  uranium,  which  under  the  influence  of 
sunlight  fluoresces  in  a  manner  not  unlike  the  fluorescence  of  the 
glass  tube  under  the  influence  of  the  cathode  rays,  might  also  give 
rise  to  X  rays  when  exposed  to  sunlight.  He  therefore  wrapped 
up  a  photographic  plate  in  black  paper,  placed  a  uranium  com- 
pound above  it,  and  exposed  the  uranium  to  sunlight.  He  found 


EADIO-ACTI VIT  Y  341 

that  after  an  exposure  of  some  days  the  photographic  plate  was 
darkened,  thus  showing  that  the  uranium  emitted  rays  which 
were  capable  of  passing  through  opaque  paper.  He  soon  found, 
however,  that  the  exposure  of  the  uranium  to  sunlight  was  quite 
unnecessary.  The  effects  could  be  produced  just  as  well  in  a  dark 
closet  as  in  sunlight.  Becquerel  therefore  drew  the  conclusion 
that  uranium  compounds  spontaneously  emit  radiations  which  are 
akin  to  X  rays  in  their  photographic  and  penetrating  effects.  He 
soon  found  also  that  these  rays  were  like  X  rays  in  their  ionizing 
properties,  but  unlike  them,  and  like  the  cathode  rays,  in  that  they 
were  deflected  both  by  magnetic  and  electrostatic  fields,  and  in  that 
they  imparted  negative  charges  to  bodies  upon  which  they  fell.  In 
a  word,  the  rays  which  produce  the  photographic  effects  observed  by 
Becquerel  were  found  to  be  in  every  respect  identical  with  cathode 
rays  except  that  the  electrons  which  are  thus  shot  off  sponta- 
neously from  uranium  atoms  have  speeds  of  about  one  half  the 
velocity  of  light.  This  is  considerably  larger  than  the  velocities 
of  ordinary  cathode  rays. 

211.  Complexity  of  the  radiations  from  uranium.  In  1899 
Rutherford  proved  clearly  that  the  rays  from  uranium  are  of  two 
kinds,  —  the  so-called  alpha  and  beta  rays.  The  beta  rays  are  simply 
projected  electrons,  that  is,  cathode  rays,  but  the  alpha  rays  have 
a  much  less  penetrating  power  and  produce  much  stronger  ioniz- 
ing effects.  They  have  since  been  shown  to  consist  of  positively 
charged  bodies  of  atomic  size,  their  mass  being  either  that  of  the 
hydrogen  molecule  or  the  helium  atom.  These  alpha  rays  are  com- 
pletely absorbed  by  a  layer  of  air  three  or  four  centimeters  thick. 
The  name  radio-activity  has  been  given  to  the  property,  found  to 
be  possessed  by  quite  a  number  of  substances,  of  spontaneously 
projecting  bodies  of  atomic  or  subatomic  size  with  such  velocity 
as  to  ionize  a  gas  or  produce  an  impression  upon  a  photographic 
plate. 

The  method  which  has  been  most  commonly  used  for  studying 
the  radiations  of  uranium,  and  of  other  substances  which  have 
been  found  to  possess  similar  properties,  is  the  following.  The 
uranium,  or  other  substance  to  be  tested,  is  placed  in  a  shallow 
metal  vessel  between  two  plates,  A  and  B  (Fig.  244),  to  one  of 


342 


ELECTRICITY,  SOUND,  AND  LIGHT 


which  is  attached  a  gold  leaf  or  other  indicator  of  an  electrical 
charge.  The  rays  from  the  radio-active  substance  ionize  the  gas 
between  the  plates,  and  if  the  plate  to  which  the  gold  leaf  is 
attached  is  originally  charged,  it  loses  this  charge  at  a  rate  which 
is  proportional  to  the  number  of  ions  formed  per  second  in  the  gas 

between  the  plates  because  of  the  pres- 
ence of  the  radio-active  substance.  The 
relative  radio-activities  of  two  substances 
are  then  compared  by  measuring  the  rel- 
ative rates  of  discharge  of  the  electro- 

scope   when    equal   weights   of   the   two 

substances  are  spread  over  equal  areas. 
When  tested  by  this  means  the  activity 
of  uranium  is  found  to  be  exceedingly 
constant;  hence  uranium  is  commonly 
taken  as  a  standard,  and  the  activities  of  all  substances  are  rated 
in  terms  of  the  activity  of  uranium. 

The  proof  first  given  by  Rutherford  of  the  complexity  of  the 
radiations  from  uranium  was  as  follows.  He  laid  layers  of  alu- 
minum foil  successively  over  the  uranium  and  observed  the  diminu- 
tion in  the  rate  of  discharge  of  the  electroscope  produced  thereby. 
The  following  table  represents  one  set  of  his  observations. 


FIG.  244 


THICKNESS  OF  ALUMINUM  FOIL  .005  MM. 


Number  of  layers 

Discharge  per  minute  in 
scale  divisions 

Ratio  of  successive  rates 
of  discharge 

0 

182 

1 

77 

42 

2 

33 

43 

3 

14.6 

44 

4 

9.4 

65 

12 

7 

It  will  be  seen  that  at  first  each  succeeding  layer  cuts  down 
the  rate  of  discharge  of  the  electrometer  by  approximately  the 
same  fraction  of  the  value  which  it  had  before  the  introduction 


EADIO-ACTIVITY  343 

of  that  layer;  in  other  words,  the  ratio  of  each  two  successive 
values  of  the  leak  is  constant.  This  is,  of  course,  as  it  should  be  if 
there  is  but  one  type  of  radiation  which  is  being  absorbed  by  suc- 
cessive layers.  But  it  will  be  seen  that  after  the  third  layer  this 
relation  breaks  down  completely,  and  the  insertion  of  eight  new 
layers  after  the  fourth  reduces  the  leak  only  from  9.4  to  7.  This 
is  interpreted  as  meaning  that  the  alpha  rays  have  been  practically 
all  absorbed  by  the  first  four  layers,  while  the  beta  rays  are  but 
very  slightly  absorbed  by  the  introduction  of  eight  more  layers.  In 
order  to  cut  the  remaining  leak  down  to  one-half  value,  i.e.  to  3.5, 
it  was  found  necessary  to  introduce  a  thickness  of  aluminum  equal 
to  100  more  sheets.  Since  one  layer  cuts  down  the  alpha  radia- 
tion to  less  than  one-half  value,  the  beta  radiation  must  be  more 
than  one  hundred  times  as  penetrating  as  the  alpha  radiation. 

The  above  example  illustrates  one  method  of  separating  the 
two  types  of  rays.  Another  method  is  to  pass  the  radiations 
through  a  strong  magnetic  field.  The  beta  rays  are  found  to  be 
easily  deflected,  while  the  alpha  rays  are  deflected  exceedingly 
little  and  in  the  opposite  direction.  It  is  this  experiment  which 
shows  that  the  two  types  of  rays  are  oppositely  charged. 

212.  Other  radio-active  substances.  Immediately  after  Bec- 
querel's  discovery  of  uranium  rays  M.  and  Mme.  Curie,  of  Paris, 
made  a  careful  study  of  all  the  then  known  elements  by  the  elec- 
trical method,  with  a  view  of  determining  whether  or  not  the 
property  of  radio-activity  was  possessed  by  other  substances  than 
uranium.  They  found  that  thorium  and  all  the  thorium  compounds 
had  about  the  same  degree  of  activity  as  uranium  and  its  compounds, 
but  that  no  other  of  the  then  known  elements  behaved  in  this  way. 
However,  in  investigating  the  ores  of  uranium  they  found  that 
pitchblende,  which  is  composed  largely  of  uranium  oxide,  had  about 
four  times  the  activity  of  chemically  prepared  uranium  oxide. 
Since,  in  general,  the  chemically  prepared  compounds  of  uranium 
were  found  to  have  activities  which  were  proportional  to  the 
amounts  of  uranium  present,  it  appeared  clear  that  the  activity  of 
uranium  was  a  property  of  the  uranium  atom  and  was  not  affected 
by  the  sort  of  combination  in  which  the  atom  was  found.  From 
these  facts  the  Curies  concluded  that  the  large  activity  of 


344  ELECTRICITY,  SOUND,  AND  LIGHT 

pitchblende  must  be  due  to  the  presence  of  some  hitherto  unknown 
element  of  very  great  activity.  They  at  once  applied  themselves 
to  the  task  of  separating  this  element.  Their  methods  were  those 
commonly  used  in  chemical  analysis,  except  that,  after  each  separa- 
tion by  precipitation,  both  the  filtrate  and  the  precipitate  were 
evaporated  to  dryness  and  tested  for  their  activities  by  the  elec- 
trical method.  It  was  thus  possible  to  find  whether  the  active 
substance  had  been  precipitated  out  of  the  solution,  or  whether  it 
had  remained  in  the  filtrate.  The  result  of  this  search  was  the  defi- 
nite separation  from  a  ton  of  pitchblende  of  a  few  centigrams  of  a 
new  element  which  was  named  radium  because  it  had  the  tremen- 
dous radio-activity  of  about  two  million  as  compared  with  uranium. 

In  chemical  properties  radium  is  closely  allied  to  barium,  from 
which  it  was  found  very  difficult  to  separate  it.  Another  radio- 
active substance  was  later  extracted  from  pitchblende  and  named 
actinium.  This  substance  is  chemically  allied  to  thorium. 

The  radiations  from  radium  and  actinium,  like  those  from 
uranium,  have  been  found  to  consist  of  alpha  and  beta  rays.  A 
third  type  of  radiation,  however,  called  the  gamma  rays,  was  also 
found  to  be  emitted  by  radium  and  actinium.  This  is  character- 
ized by  a  penetrating  power  a  hundred  times  greater  than  that  of 
the  beta  rays  and  by  its  entire  freedom  from  deflectability  by  elec- 
trostatic or  magnetic  fields.  These  gamma  rays  are  commonly 
supposed  to  be  X  rays,  which  in  any  case,  according  to  the  theory 
of  section  209,  would  of  necessity  accompany  the  beta  rays  as  they 
do  in  a  cathode-ray  tube.  In  the  case  of  uranium  also  these  gamma 
rays  are  doubtless  present,  but  are  so  weak  as  to  escape  detection 
under  the  ordinary  conditions  of  experiment. 

213.  The  spinthariscope.  The  most  striking,  though  perhaps 
not  the  most  conclusive,  evidence  that  radio-active  substances  are 
continually  projecting  particles  from  their  atoms  with  enormous 
velocity  is  furnished  by  an  experiment  devised  by  Crookes  in  1903. 
When  a  bit  of  radium  is  placed  near  a  screen  of  zinc  sulphide  it 
produces  fluorescence  in  the  screen  precisely  as  do  cathode  rays  or 
X  rays.  In  order  to  study  more  carefully  this  fluorescence  Crookes 
placed  a  tiny  speck  of  radium  R  (Fig.  245)  about  a  millimeter 
above  a  zinc  sulphide  screen  $  and  then  viewed  the  latter  through 


R 


RADIO-ACTIVITY  345 

a  lens  L  which  produced  from  10  to  20  diameters  magnification. 
The  continuous  soft  glow  of  the  screen,  which  is  all  that  one  sees 
with  the  naked  eye,  is,  under  these  circumstances,  resolved  by  the 
microscope  into  a  thousand  tiny  flashes  of  light.    The 
appearance  is  as  though  the  screen  were  being  vigor- 
ously bombarded  by  a  continuous  rain  of  projectiles, 
each  impact  being  marked  by  a  flash  of  light.    The 
flashes  are  probably  due  indirectly  to  the  impacts  of 
the  alpha  particles.    Since  these  alpha  particles  have 
been  shown  by  measurements  upon  their  electrostatic 
and  magnetic  deflectability  to  have  about  the  mass  of 
a  helium  atom,  and  to  have  velocities  of  about  one  tenth  that  of 
light,  it  will  be  obvious  that  they  would  strike  an  object  with  tre- 
mendous energy. 

214.  Radio-activity  an  atomic  property.    Whatever  the  cause 
of  this  ceaseless  emission  of  particles  by  radio-active  substances,  it 
is  now  fairly  well  established  that  the  activity  of  all  radio-active 
substances  is  proportional  to  the  amount  of  the  radio-active  ele- 
ment present,  and  has  nothing  whatever  to  do  with  the  nature  of 
the  chemical  compound  in  which  the  active  element  is  found. 
Thus  uranium,  for  example,  may  be  changed  from  a  nitrate  to  a 
chloride,  or  from  a  chloride  to  a  sulphide  or  an  oxide,  without  pro- 
ducing any  change  whatever  in  its  activity.    Furthermore,  radio- 
activity has  been  found  to  be  equally  independent  of  all  changes 
in  physical  as  well  as  chemical  condition.  The  activity  of  uranium, 
for  example,  is  not  altered  at  all  by  reducing  its  temperature  to 
that  of  liquid  air.    This  is  strong  evidence  in  favor  of  the  view 
that  radio-active  change,  that  is,  the  change,  whatever  it  be,  which 
is  responsible  for  the  emission  of  the  alpha  and  beta  particles, 
involves  a  change  in  the  nature  of  the  atom  itself. 

215.  Radio-active  transformations.    The  first  direct  evidence 
that  the  atoms  of  radio-active  substances  are  continually  trans- 
muting themselves  into  atoms  of  new  physical  and  chemical  prop- 
erties was  contained  in  some  experiments  made  in  1900  by  Sir 
William  Crookes,  but  first  interpreted  by  Eutherf ord.    Crookes  found 
that  when  he  precipitated  an  aqueous  solution  of  uranium  nitrate 
with  ammonium  carbonate  and  then  redissolved  the  precipitated 


346  ELECTEICITY,  SOUND,  AND  LIGHT 

uranium  by  an  excess  of  the  reagent,  there  remained  behind  an 
undissolved  precipitate  which  contained  a  large  part  of  the  original 
activity  possessed  by  the  uranium  nitrate.  He  called  this  undis- 
solved precipitate  (or  better,  the  portion  of  it  which  was  responsi- 
ble for  the  activity,  for  when  chemically  tested  it  showed  nothing 
but  iron,  aluminum,  and  other  impurities)  uranium  X.  He  soon 
afterwards  discovered  that  the  uranium  nitrate,  which  had  lost  a 
large  part  of  its  activity  through  the  separation  from  it  of  uranium 
X,  had,  in  the  course  of  a  few  months,  completely  regained  its 
original  activity,  while  the  uranium  X  had  lost  its  power  of  radi- 
ating. This  result  pointed  very  strongly  to  the  conclusion  that  a 
radio-active  substance  is  being  continually  produced  by  uranium. 
Not  long  after  this  Eutherford  found  that  a  product  analogous  to 
uranium  X  could  be  separated  from  thorium  by  a  similar  method. 
He  called  this  thorium  X.  The  essential  difference  between  ura- 
nium X  and  thorium  X  lies  in  the  fact  that  the  activity  of  the 
first  falls  to  half  value  in  twenty-two  days,  while  that  of  the 
second  falls  to  half  value  in  about  four  days. 

The  examination  of  radium  revealed  a  behavior  exactly  similar 
to  that  of  uranium,  for  it,  too,  was  found  to  be  continually  produc- 
ing a  radio-active  substance  which,  when  separated  from  the  radium, 
slowly  lost  its  activity,  while  the  radium  from  which  it  was  sepa- 
rated regained,  at  a  like  rate,  its  radiating  power.  In  the  case  of 
radium  this  new  substance,  unlike  uranium  X  and  thorium  X, 
could  be  distinguished  by  other  physical  properties  besides  its 
activity.  Thus  Rutherford  found  it  to  be  of  the  nature  of  a  gas 
which  could  be  separated  from  a  radium  salt  by  heating  the  latter 
or  by  dissolving  it  in  water.  The  radium  which  had  been  so 
treated  lost,  for  the  time  being,  all  but  one  fourth  of  its  original 
radiating  power,  the  other  three  fourths  being  found  in  the  gas,  or 
emanation,  as  Rutherford  called  it.  This  gas  could  be  set  away 
in  bottles  and  the  change  in  its  activity  watched  from  day  to  day. 
It  could  be  condensed  by  passage  through  tubes  immersed  in 
liquid  air,  its  presence  being,  in  general,  detected  by  the  ionization 
which  it  imparted  to  the  air  with  which  it  was  mixed.  It  has 
recently  been  obtained  in  sufficient  quantities  to  show  a  charac- 
teristic spectrum  and  other  qualities  common  to  gases.  This  gas, 


RADIO-ACTIVITY  347 

however,  like  uranium  X  and  thorium  X,  has  but  a  transitory 
existence,  for  the  fact  that  it  gradually  loses  its  activity  shows 
that  it  passes  on  into  something  else. 

Nor  did  physicists  have  long  to  look  in  order  to  discover  this 
substance  into  which  the  emanation  from  radium  is  transformed. 
They  found  that  when  the  gas  comes  into  contact  with  a  solid 
object,  this  object,  especially  if  it  is  negatively  charged,  becomes 
coated  with  a  film  of  radio-active  matter  which  can  be  dissolved 
with  hydrochloric  or  sulphuric  acid,  and  which  is  left  in  the  dish 
when  the  acid  is  evaporated,  or  which  may  be  rubbed  off  with 
leather  and  found,  by  means  of  the  property  of  activity  which  it 
possesses,  in  the  ash  of  the  leather  after  the  leather  has  been 
burned.  This  active  deposit,  generally  obtained  by  placing  a  nega- 
tively charged  wire  in  a  vessel  containing  radium,  is  so  infinitesi- 
mal in  amount  that  it  has  never  been  detected  in  any  other  way 
than  through  its  activity.  At  first  it  might  look  as  though  it  were 
nothing  but  the  active  gas  itself  condensed  on  the  surface  of  the 
solid  object,  but  since  the  rate  at  which  it  loses  its  activity  is  alto- 
gether different  from  the  rate  at  which  the  emanation  decays ;  since, 
furthermore,  the  emanation  atom  is  not  positively  charged,  and 
therefore  does  not  tend  to  collect  on  a  negative  wire;  and  since 
the  active  deposit  is  found  to  emit  both  alpha  and  beta  rays,  while 
the  emanation  emits  only  alpha  rays,  it  seems  necessary  to  con- 
clude that  this  film  of  active  matter  is  the  product  of  the  emana- 
tion rather  than  the  emanation  itself.  In  fact,  it  appears  to  bear  in 
all  respects  the  same  relation  to  the  emanation  which  the  emana- 
tion bears  to  radium;  that  is,  'it -is  the  result  of  the  disintegration 
of  the  atom  of  the  emanation,  just  as  the  emanation  is  the  result 
of  the  disintegration  of  the  atom  of  radium. 

Thorium  is  found  to  be  precisely  like  radium  in  that  it  gives 
rise  to  a  gaseous  emanation,  and  in  that  this  emanation  disinte- 
grates into  something  else  which  collects  upon  a  negatively  charged 
wire.  It  has  been  definitely  shown,  however,  that  the  thorium 
emanation  is  not  the  direct  product  of  thorium,  but  rather  of 
thorium  X. 

In  order  to  collect  the  active  deposit  from  the  thorium  emana- 
tion it  is  only  necessary  to  thrust  a  wire  through  a  cork  which 


348 


ELECTRICITY,  SOUND,  AND  LIGHT 


Hi 

m 

THORIUM 

OXIDE 

—  iiii~ 

—  llll 

+ 

FIG.  246 


closes  a  small  tin  can  in  which  some  thorium  oxide  is  placed,  and 
make  the  wire  the  negative  terminal  of  a  battery  of  100  volts  or 
more,  the  can  being  made  the  positive  terminal  (see  Fig.  246).*  In 
the  course  of  an  hour  or  more  the  wire  will  have  become  appre- 
ciably active. 

The  fact  that  this  active  mate- 
rial is  deposited  on  a  negatively 
charged  wire  throws  some  light 
on  the  probable  nature  of  the  dis- 
integration of  the  emanation  atom, 
for  this  means,  of  course,  that  the 
atoms  which  are  thus  deposited  are 
positively  charged.  In  order  to  account  for  this  we  must  assume, 
that  although  the  emanation  atom  produces  only  alpha  rays,  yet 
it  must  lose  one  or  more  electrons  in  passing  over  into  the  atom 
of  this  active  deposit.  The  reason  that  these  electrons  do  not 
produce  beta-ray  effects  may  be  that  they  are  projected  with 
insufficient  velocities  to  ionize  the  air. 

If  a  wire  is  exposed  to  the  thorium  emanation  for  several  days 
in  the  manner  indicated  above,  upon  removal  its  activity  is  found 
to  decay  fairly  regularly,  falling 
to  half  value  in  about  1 1  hours, 
but  if  it  is  exposed  to  a  large 
amount  of  the  emanation  for 
only  a  few  minutes,  its  activity 
is  found  to  increase  regularly 
for  about  3J-  hours  and  then  to 

2 

decrease  in  the  manner  indi- 
cated in  Figure  247.  Jhe  in- 
itial increase  in  activity  shows 
that  an  active  substance  must  pIG-  247 

be  forming  on  the  wire  for  some 

time  after  its  removal  from  the  vessel  containing  thorium,  and 
the  fact  that  the  initial  activity  of  the  wire  is  practically  zero 
shows  that  the  material  from  which  this  active  substance  is 


100 

^ 

^ 

~~^~~ 

ACTIV/T) 

*  3  J 

/ 

/ 

/ 

I 

o 

1 

4-0        8O       120      160     2.OO    24-O    26O     32 
TIME    IN  MINUTES 

*  Figures  246  and  247  are  taken  from  Rutherford's  "Radio-active  Trans- 
formations." 


RADIO-ACTIVITY 


349 


Uranium     Ur.  X 
GxlOByr.        22  dv. 


formed  is  not  itself  active.  The  curve  is  completely  explained 
if  we  assume  that  the  atom  of  the  thorium  emanation  disinte- 
grates into  the  atom  of  some  nonactive  substance  called  by 
Rutherford  thorium  A,  which  deposits  upon  the  wire,  while  this 
nonactive  substance  then  disintegrates  into  an  active  substance, 
fc.  „  thorium  B. 

Enough  has  been  said  to  show  how,  one 
after  another,  the  successive  products  of  radio- 
active change  have  been  discovered  and  studied. 
The  following  chart  (Fig.  248),  taken  from 

Rutherford's  "  Radio- 
active Transformations," 
shows  the  various  trans- 
formations which  the 
radio-active  elements 
are  supposed  to  un- 
dergo, the  time  given 
beneath  each  product 
representing  the  time 
required  for  this  sub- 
stance to  lose  half  its 
Act.  A  Act.  B 
36  min.  2  min.  activity.  It  IS  nOW 

""acdveTi^Tr'  pretty  well  established, 


«"  .*  ^ 

o-o-d-o- 


Thorium    Th.  X 
24  x  109  yr<       4  dy. 


Eman. 
54  sec. 


Th.  A        Th.  B 

11  lir.  55  min. 


active  deposit 


.    A 


Actinium  Act.  X 
10  dy. 


Eman. 

4  sec. 


Radium     Eman. 
1300  yr.          4  dy. 


Rad.  A.     Rad.  B. 

3  mill.    .      21  min. 


Rad.  C.     Rad.  D.     Rad.  E.     Rad.  F. 


active  deposit,  rapid  change 
EIG.  248 


28  min.          40  yr. 

Radio-Lead 


6  dy. 


143  dy. 
Polonium 


active  deposit,  slow  change 


however,  that  radium  is  itself  only  a  transformation  product  of 
uranium,  being  produced  not  by  the  immediate  disintegration 
of  uranium  X,  but  by  some  rayless  change  in  an  intermediate 
product  called  ionium  into  which  the  uranium  X  immediately  dis- 
integrates. It  seems  probable,  too,  that  actinium  is  also  a  disinte- 
gration product  of  uranium.  If  this  should  turn  out  to  be  the  case, 


350  ELECTRICITY,  SOUND,  AND  LIGHT 

there  would  be  but  two  strongly  radio-active  families,  namely  the 
uranium  family  and  the  thorium  family.  It  is  worthy  of  remark 
that  the  atoms  of  uranium  and  thorium  are  the  heaviest  atoms 
known,  their  atomic  weights  being  239  and  232  respectively. 
These  atoms,  then,  seem  to  be  continually  disintegrating  into  atoms 
of  lower  atomic  weight,  the  process  of  disintegration  apparently 
consisting  in  the  expulsion  of  an  alpha  or  a  beta  particle,  or  of 
both  together.  The  residue  of  the  atom  after  each  such  expulsion 
appears  to  have  new  physical  and  chemical  properties,  that  is,  to 
be  a  new  chemical  substance.  This  new  substance  is  itself  in  geii- 

o 

eral  unstable,  and  after  a  time  passes  over  into  something  else, 
with  the  expulsion  of  other  particles.  What  are  the  ultimate 
products  of  this  series  of  radio-active  changes  is. not  yet  definitely 
known.  It  is  certain  that  helium  is  one  of  them,  for  the  spectrum 
of  helium  has  been  found  to  grow  out  of  the  emanation  of  radium. 
Rutherford  -regards  the  alpha  particles  as  themselves  helium  atoms. 
It  is  not  impossible  that  lead  and  some  other  common  elements 
are  products  of  this  disintegration.  At  any  rate,  many  of  these 
elements  are  regularly  found  in  uranium  ores.  J.  J.  Thomson  has 
recently  (1907)  shown  that  the  same  alpha  particle  which  is  pro- 
jected from  radio-active  substances  is  thrown  off  from  all  sorts  of 
substances  in  a  highly  exhausted  tube  under  the  influence  of  very 
intense  electrical  fields.  If  any  of  these  common  substances  are 
spontaneously  emitting  these  alpha  particles  at  a  very  slow  rate, 
and  with  velocities  too  small  to  ionize  the  surrounding  air,  we 
should  have  no  means  of  detecting  the  fact.  Whether,  then,  the 
continuous  transmutation  of  one  element  into  another  is  confined 
to  a  few  radio-active  substances,  or  whether  it  is  a  general  phe- 
nomenon of  nature,  is  a  question  which  must  be  left  for  the 
future  to  decide. 

EXPERIMENT  28 

(A)  Object.  To  compare  the  radio-activities  of  black  uranium  oxide, 
pitchblende,  thorium  nitrate,  and  uranium  nitrate. 

Directions.  With  a  mortar  and  pestle  reduce  to  a  fluffy  powder  each  of 
the  substances  to  be  compared.  Weigh  out,  say,  4  g.  of  each,  and  spread 
uniformly  over  a  metal  surface  of  about  25  sq.  cm.  This  may  be  done 
either  by  pressing  down  the  powder  with  some  flat  object  in  a  shallow 


RADIO-ACTIVITY 


351 


vessel,  of,  say,  3  mm.  depth,  until  the  surface  is  smooth  and  the  thickness 
uniform,  or  by  making  an  emulsion,  in  a  small  test  tube,  of  the  powder 
in  chloroform,  alcohol,  or  some  other  liquid  in  which  the  powder  will 
not  dissolve,  and  then  pouring  the  emulsion  very  quickly  into  the  shallow 
vessel  and  setting  away  until  the  liquid  has  evaporated. 

Charge  the  electroscope  (Fig.  249)  by  drawing  a  charged  rod  of  sealing 
wax  over  the  projecting  end  e  of  the  metal  rod  which  supports  the  gold 
leaf,  and  which  is  insulated  from  the  metal  frame  E  of  the  electroscope 
by  means  of  an  amber  plug  p. 

When  a  deflection  of  from   30°  to  45°  has  been  produced,  focus  the 
telescope,   or  microscope,    upon  the   gold  leaf   and   set    the    eyepiece    so 
that    the    leaf    in    discharging   moves    as    nearly 
as   possible  at  right  angles  to  the   scale  in  the 
eyepiece. 

Insert  the  uranium  oxide  into  the  electroscope 
through  one  of  the  glass  sides,  and  earth  the  metal 
case  E  by  running  a  small  wire  from  it  to  a  gas 
or  water  pipe.  Place  the  cylindrical  metal  tube  b 
over  e  so  as  to  cut  off  all  outside  inductive  effects 
(see  p.  12),  and  make  two  or 
three  observations  with  a  stop 
watch  of  the  time  required  for 
the  leaf  to  pass  over  a  given 
five  divisions  of  the  scale  in  the 
eyepiece,  recharging  the  leaf 
with  the  sealing  wax  between 
each  set  of  readings.  Replace 
the  uranium  by  the  other  sub- 
stances to  be  compared,  being 
careful  to  place  them  all  in 
just  the  same  position,  and 
make  similar  sets  of  readings 
for  each.  Express  the  activities  of  all  the  substances  in  terms  of  that  of 
the  black  uranium  oxide. 

(B)  Object.  To  determine  the  number  of  ions  produced  per  second  in  a 
gas  by  the  radiations  from  a  uranium  oxide  film  of  a  given  weight  per 
square  centimeter. 

Directions.  Replace  the  uranium  oxide  film  in  the  electroscope,  but  this 
time  screw  upon  e  the  solid  brass  cylinder  a,  and  set  over  this,  as  coaxially 
as  possible,  the  brass  tube  b.  This  operation  places  a  condenser  in  parallel 
with  the  electroscope,  so  that  the  capacity  of  the  system  is  greatly  increased. 
Since  the  uranium  produces  the  same  number  of  ions  per  second  as  before, 
the  same  charge  will  be  taken  per  second  from  the  electroscope,  but  its 
capacity  being  now  much  increased,  the  rate  of  fall  of  the  leaves  will  be 


FIG.  249 


352 


ELECTRICITY,  SOUND,  AND  LIGHT 


much  reduced.  Let  tl  and  /2  represent  the  two  times  of  fall  of  the  gold 
leaf  between  the  given  limits  before  and  after  the  condenser  is  added  ;  let 
PD  represent  the  fall  in  potential  corresponding  to  the  observed  fall  of  the 
leaf  ;  let  c  be  the  capacity  of  the  electroscope  alone,  and  C  the  capacity  of 
the  condenser  ;  let  Qt  and  Q2  be  the  charges  removed  from  the  electroscope 
in  the  times  t1  and  t2  respectively. 

Then  =  1 


But  by  Chapter  VIII,     Q1  =  PD  x  c,  and  Q2  =  PD  x  (c+C). 


Hence 


This  gives  the  capacity  of  the  electroscope  in  terms  of  tv  and  £2,  and  the 
capacity  C  of  the  condenser.  This  last  quantity  may  be  computed  from 
the  mean  area  of  the  inner  and  outer  cylindrical  surfaces  and  their  distance 

apart*  (see  eq.  17,  p.  109).  If  now  we  can  ob- 
tain the  P.D.  corresponding  to  the  observed 
change  in  deflection  of  the  gold  leaf,  we  have 
at  once  the  quantity  of  electricity  passing  per 
second  to  the  gold-leaf  system.  Calling  this 
quantity  i,  we  have 

.=  C1  =  P5xo 

' 


To  obtain  PD  in  volts,  attach  the  electro- 
scope in  parallel  with  a  Braun  electrometer 
(Fig.  250),  or  other  electrostatic  voltmeter, 
and  charging  both  by  means  of  the  sealing 
wax,  find  by  direct  comparison  the  P.D.  cor- 
responding to  the  observed  change  in  deflec- 
tion of  the  gold  leaf.  Divide  this  by  300 

(i.e.  —          j ,  to  reduce  it  to  absolute  elec- 

FIG.  250 

trostatic  units.    Express  this  current  also  in 

electro-magnetic  units  by  dividing  by  3  x  1010.    The  charge  on  each  ion  of 
an  ionized  gas  is  known  to  be  the  same  as  the  charge  on  the  hydrogen  ion 

*  The  rigorous  formula  for  a  cylindrical  condenser  of  this  kind  is  C  = •> 


in  which  I  is  the  length  of  the  inner  cylinder,  a  the  external  diameter  of  this 
cylinder,  and  b  the  internal  diameter  of  the  outer  tube.  This  is  not  appreciably 
different  from  the  above  unless  a  and  6  differ  considerably. 


EADIO-ACTIVITY  353 

in  electrolysis,  namely  about  4  x  1Q-10  absolute  electrostatic  units.    Hence 
the  number  of  positive  ions  produced  per  second  by  a  square  centimeter  of 

the  uranium  film  is  simply  -  —  • 
J  4  xlO-10 

EXAMPLE 

(A)  When  4  g.  each  of  uranium  oxide,  pitchblende,  uranium  nitrate, 
and  thorium  nitrate  were  spread  uniformly  over  circular  vessels  of  5.5  cm. 
diameter,  and  having  rims  3  mm.  in  height,  the  observations  on  the  times 
required  for  the  leaf  to  fall  from  division  5  to  division  10  were  : 

1st  2d  3d  Mean 

Uranium  oxide      ....  19.6  19.8  19.4  19.6 

Pitchblende      .....  6.4  6.4  6.6  6.5 

Uranium  nitrate    ....  54.6  54.8  54.6  54.7 

Thorium  nitrate    ....  16.4  16.6  16.8  16.6 

In  terms  of  the  uranium  oxide  the  activity  of  the  pitchblende  was 
therefore  3.17,  that  of  the  uranium  nitrate  .36,  and  that  of  the  thorium 
nitrate  1.18. 

(B)  When  a  cylindrical  .condenser  5.89  cm.  long,  the  diameter  of  whose 
inner  and  outer  cylindrical  surfaces  were  1.594  cm.  and  1.90  cm.  respectively, 
was  added,  the  mean  time  of  fall  of  the  leaf  through  5  divisions  when  the 
uranium  oxide  was  in  the  electroscope  was  123.4  seconds.    The  capacity  C 
was  found  to  be  16.8  absolute  electrostatic  units.    Hence  c  =  3.19  electro- 
static units.    The  loss  of  PD  due  to  fall  of  the  gold  leaf  through  the  given 
5  divisions  was  68  volts.    The  uranium  oxide  was  spread  over  a  circular  area 

.      Q      PDxc       68  x  3.19  ft  • 

having  a  diameter  ot  o.o  cm.    Hence  i  —  —  —  —        —  —  -  -  --  =  .Ooo9 

*  <  300  x  19,6 


electrostatic  units*  and  i  per  sq.  cm.  =  —  -  -  =  .00155  electrostatic  units 


AO  £*Q 

=  5.  17  x  10~14  electro-magnetic  units.   Therefore  the  number  of  ions  formed 
per  second  by  the  radiation  from  1  sq.  cm.  =  3,900,000. 


PROBLEMS 


CHAPTERS  I  AND  II 

1.  Find  the  intensity  of  the  magnetic  field  due  to  an  isolated  south  pole 
of  560  units  strength  at  a  point  20  cm.  from  the  pole.    If  the  value  of  //, 
the  horizontal  component  of  the  earth's  field  is  0.24  in  this  neighborhood, 
what  will  be  the  resultant  field  intensity  and  its  direction  if  the  point  con- 
sidered is  due  magnetic  north  of  the  pole  ? 

2.  A  magnet  has  a  length  of  10  cm.  and  a  pole  strength  of  450  units. 
Calculate  the  direction  of  the  field  with  respect  to  the  axis  of  the  magnet, 
and  the  intensity  at  a  point  20  cm.  from  one  pole  and  15  cm.  from  the  other. 

3.  What  is  the  field  intensity  due  to  a  magnet  of  moment  M  at  a  point 
distant  r  cm.  along  a  perpendicular  to  the  magnet  at  its  middle  point? 

4.  If  a  magnet  vibrates  with  a  period  of  5  sec.  where  the  horizontal 
intensity  is  0.18,  what  is  its  period  where  the  intensity  is  0.24? 

5.  A  very  short  magnetic  needle  is  suspended  17.32  cm.  below  the  cen- 
ter of  a  bar  magnet  of  pole  strength  200  units.    The  length  of  the  bar 
magnet  is  20  cm.  between  poles.    Its  axis  (S  to  N)  makes  an  angle  of  90° 
with  the  direction  of  the  horizontal  component  (H  =  0.24)  of  the  earth's 
field.    What  angle  does  the  axis  of  the  magnetic  needle  make  with  H 
when  in  equilibrium  ? 

6.  If  the  period  of  oscillation  of  the  magnetic  needle  in  Problem  5  is 
3  sec.,  what  is  it  when  the  bar  magnet  has  been  rotated  through  90°? 

7.  A  magnetic  needle  is  suspended  above  and  parallel  to  a  horizontal 
bar  magnet  lying  in  the  magnetic  meridian.    When  the  north  end  of  the 
bar  points  northward  the  period  of  the  magnet  is  7  sec.,  but  when  the  bar 
magnet  is  reversed  the  period  is  5  sec.    What  would  be  the  period  in  the 
earth's  field  alone  ? 

8.  A  bar  magnet  suspended  to  rotate  about  a  vertical  axis  has  a  period 
of  4  sec.  in  the  earth's  field.    Find  the  period  of  the  system  if  a  bar  of 
aluminum  is  attached  to  the  magnet  so  as  to  rotate  with  it  about  the  same 
axis,  the  aluminum  bar  having  one  half  the  length  and  one  half  the  width 
of  the  magnet,  but  the  same  vertical  thickness.    The  density  of  steel  is 
7.7  and  of  aluminum  2.7.    Find  also  the  value  of  the  moment  of  inertia  of 
the  bar  if  the  magnetic  moment  is  1200  units  and  H  is  0.22. 

354 


PROBLEMS  355 

CHAPTER  III 

9.  What  current  must  be  passed  through  a  tangent  galvanometer  coil 
45  cm.  in  diameter  and  consisting  of  5  turns,  in  order  that  the  needle  may 
be  deflected  30°  in  a  neighborhood  where  H=  0.18?  What  current  for  45° 
and  for  60°? 

10.  What  must  be  the  radius  of  a  single  coil  which,  when  carrying  a 
given  current,  would  produce  at  its  center  a  field  intensity  equal  to  that 
caused  by  the  same  current  in  passing  through  two  concentric  coaxial  coils 
of  15  cm.  and  45  cm.  diameters  ? 

11.  A  circular  wire  coil  of  15  turns  and  50  cm.  diameter  must  carry 
how  much  current  to  produce  at  its  center  a  field  of  0.60? 

12.  A   current  of    0.331    ampere    is    passed    through    two    concentric, 
coaxial   coils  of    12   and  24   turns   respectively,  first  when  connected   so 
that  their  magnetic  fields  are  in  opposite  directions,  and  second  so  that 
they  are  in  the  same  direction.    The  fields  thus  established  are  0.20  and 
0.30  respectively.     Find  the  radii  of  the  two  coils. 

13.  Draw  the  curve  representing  the  values  of  the  tangent  for  all  angles 
between  0°  and  90°.    Decide  from  a  study  of  the  curve  for  what  value  of 
the  deflection  in  a  tangent  galvanometer  the  smallest  error  is  introduced 
into  the  final   result  by   a  constant   observational  error  in  reading  the 
deflection. 

14.  The  reading  of  a  tangent  galvanometer  was  46°  unconnected  for  tor- 
sion.   When  the  torsion  head  was  twisted  through  that  angle,  the  needle 
turned  1.1°  against  the  restoring  force  of  the  earth's  field.    Find  the  per 
cent  of  error  introduced  into  the  final  result  by  using  the  uncorrected 
instead  of  the  corrected  value  of  the  deflection. 

15.  A  current  which  gives  a  reading  of  0.27  ampere  on  a  milliammeter 
deposits  0.2008  g.  of  silver  in  10  min.  42  sec.    What  is  the  error  in  the 
ammeter  reading  ? 

16.  How  much  water  should  be  decomposed  by  a  current  of  0.80  ampere 
in  one  hour? 

17.  Given  the  electro-chemical  equivalent  of  zinc  as  0.03367  g.  per  cou- 
lomb, find  how  much  zinc  is  consumed  in  a  Daniell  cell  in  generating  a 
current  of  0.5  ampere  for  150  hr.    How  much  copper  is  deposited  under 
the  conditions  of  the  problem  ? 

18.  A  current  is  sent  through  three  cells,  one  containing  acidulated 
water,   the   second   copper   sulphate,    and  the  third   silver   nitrate.    How 
much  copper  will  have  been  deposited  in  the  second  cell  while  2  g.  of 
silver  are  deposited  in  the  third  cell?    What  will  be  the  volume,  at  76  cm. 
of  mercury  pressure  and  20°  C.,  of  the  mixed  gases  liberated  in  the  first  cell? 


356  ELECTRICITY,  SOUND,  AND  LIGHT 

19.  How  many  ounces  of  aluminum  are  deposited  from  a  suitable  solu- 
tion of  bauxite,  a  natural  aluminum  oxide,  by  the  passage  of  150  amperes 
for  one  hour?    (Aluminum  oxide  =  A12O3  :  atomic  weight  of  Al  =  27.4.) 

20.  Hydrogen  given  off  at  an  electrode  of  a  hydrogen  voltameter  dis- 
places entirely  the  volume  of  40  cc.  of  water  contained  by  the  vessel  placed 
over  the  electrode.    The  atmospheric  pressure  is  747  mm.  of  mercury  at 
the  room  temperature  of  23°  C.    Find  the  weight  of  the  liberated  gas. 

21.  The  atomic  weights  of  three  elements  are  in,  n,  and  p.    They  all 
form  sulphates  of  the  form  M2SO4.    What  is  the  ratio  of  their  electro- 
chemical equivalents  ? 

22.  An  element  which  forms  oxides  of  the  form  E2O3  and  EO  has  an 
electro-chemical  equivalent  of  e  for  the  compound  of  higher  valency.   What 
is  the  electro-chemical  equivalent  for  the  lower  valency  ? 

CHAPTERS  IV  AND  V 

23.  When  electrical  energy  costs  8  cents  per  kilowatt  hour,  how  much 
does  it  cost  to  operate  an  incandescent  lamp  that  takes  0.5  ampere  from 
110  volt  mains  ? 

24.  A  vessel  of  30  g.  water  equivalent  contains  1600  g.  of  water.    Its 
radiation  constant  is  such  that  its  temperature  falls  at  the  rate  of  6°  C. 
per  minute  at  90°  C.     How  much  current  must  be  passed  through  a  wire 
of  10  ohms'  resistance  immersed  in  the  water  that  the  temperature  shall  be 
maintained  at  90°  C.  ? 

25.  From  Experiment  4  it  is  seen  that  the  rate  at  which  heat  is  devel- 
oped in  a  wire  is  (PD)L    Show  that  this  is  also  equal  to  RIZ. 

26.  A  wire  immersed  in  water  generates  heat  at  the  rate  of  2  calories 
per  second  when  carrying  0.5  ampere  of  current.    Find  the  power  in  watts 
and  in  horse  power  expended  in  the  wire  and  its  resistance. 

27.  Show  from  Ohm's  law  that  in  two  wires  of  the  same  circuit  the 
ratio  of  the  P.D.'s  between  the  terminals  of  the  wires  is  the  same  as  the 
ratio  of  their  resistances. 

28.  Neglecting  the  loss  due  to  radiation,  what  will  be  the  rise  in  tem- 
perature in  10  sec.  of  a  No.  20  copper  wire  of  mass  45.5  g.  and  resistance 
0.3293  ohm,  connected  across  constant  potential  mains  of  25  volts  P.D.? 

29.  If  a  wire  of  the  same  mass  as  that  in  Problem  28,  but  of  half  the 
diameter,  is  supplied  with  current  from  the  same  mains,  what  will  be  its 
rise  in  temperature  in  10  sec.? 

30.  If  the  wire  is  of  the  same  diameter  as  that  in  Problem  28,  but  of 
half  the  length,  and  hence  half  the  mass,  what  will  be  its  rise  in  10  sec.? 


PROBLEMS 


357 


FIG.  1 


31.  If  equal  lengths  of  wires  of  different  diameters  but  of  the  same 
material  are  placed  across  constant  potential  mains,  which  will  burn  out 
the  sooner,  the  small  wire  or  the  large  one?    In  answering  this  question 
consider  radiation. 

32.  If  the  ammeter  in 
Figure  1   reads  0.5  am- 
pere,   the    voltmeter    50 
volts,  and  if   the  resist- 
ance of  the  voltmeter  is 
300   ohms,   what  is    the 
resistance  of  the  coil  R  ? 

33.  Three  coils  of  re- 
sistance 2,  3,  and  5  ohms 
are    connected    in    series 

across  110  volt  mains.    Find  the  current  through  the  circuit,  the  P.D. 
across  the  terminals  of  each  coil,  and  the  power  expended  in  each  coil. 

34.  A  2-  and  a  7 -ohm  coil  are  part  of  a  circuit  in  which  18  amperes  are 
flowing.    Calculate  the  P.D.  across  the  terminals  of  this  parallel  circuit, 
the  combined  resistance  of  the  two  coils,  and  the  current  in  each  coil. 

35.  A  room  is  lighted  by  a  chandelier  containing  11  incandescent  lamps 
in  parallel  between  wires  of  resistance  negligible  as  compared  to  the  lamp 
resistances.    Each  lamp  has  a  resistance  of  220  ohms.    What  is  the  com- 
bined resistance  which  they  offer '? 

36.  A  P.D.  of  1.06  volts  is  maintained 
at  the  terminals  of  a  battery  E  in  Fig- 
ure 2.     If    resistances    of    10,000,    1100, 
900,  and  1  ohms  are  connected  at  a,  ft, 
c,  and  d  respectively,  what  is  the  current 
through  the  resistance  c  ? 

37.  Given  the  resistance  of  the  volt- 
meter   used  in  Experiment  4   as   16,000 
ohms,  and  the  resistance  of  the  coil  of 
platinum  wire  as  13.4  ohms,  find  the  per 

cent  of  error  introduced  in  assuming  that  all  the  current  indicated  by  an 
ammeter  reading  of  4.08  amperes  passes  through  the  coil. 


FIG.  2 


CHAPTERS   VI   AND   VII 

38.  One  mil  is  a  thousandth  of  an  inch.  One  circular  mil  is  the  area  of 
a  circle  one  mil  in  diameter.  The  area  of  a  circle  d  mils  in  diameter  is 
therefore  d2  circular  mils.  The  resistance  of  pure  soft  copper  such  as  is 
used  in  making  insulated  wires  is  given  as  10.38  ohms  per  mil-foot  at 
75°  F.  What  in  C.G.S.  units  is  the  specific  resistance  of  such  copper? 


358  ELECTRICITY,  SOUND,  AND  LIGHT 

39.  What  is  the  resistance  at  75°  F.  of  1000  feet  of  wire  31.96  mils  in 
diameter  ? 

40.  The  resistance  of  aluminum  wire  is  given  as  17.03  ohms  per  mil- 
foot  at  75°  F.    What  is  the  ratio  of  the  diameters  of  two  wires,  one  cop- 
per and  the  other  aluminum,  such  that  for  the  same  length  the  resistances 
are  the  same  ? 

41.  The  specific  resistance  of  copper  at  0°C.  is  1650  absolute  C.G.S. 
units.    Find  the  resistance  of  a  trolley  wire  one  kilometer  long  and  one 
centimeter  in  diameter. 

42.  If  in  Figure  62,  page  82,  JR1  =  10,000,  Rz  =  100,  R3  =  100,  G  =  100, 
and  V  —  1.06,  what  current  was  flowing  through  (7? 

43.  The  mean  temperature  coefficient  of  copper  is  not  exactly  0.0042. 
The  mean  rate  of   increase  of   resistance  for  Matthiessen's  pure  copper 
between  0°  and  l°C.  is  best  expressed  as  a  =  (a  +  bt  +  c/2)  where  t  is  the 
temperature  entering  into  the  relation  Rt  —  -K0(l  -f  af).    The  constants  a 
and  b  have  the  values  0.004019  and  0.00000214  respectively,  for  values  of  t 
between  0°  and  100°.    Find  the  value  of  a  to  use  at  25°  C.  and  at  50° C. 

44.  Compute  the  error  introduced  in  finding  the  value  of  a  resistance 
at  50°  from  an  observation  of  its  resistance  at  25°  by  using  the  average 
value  a  —  0.0042  instead  of  the  value  of  a  as  given  in  Problem  43. 

45.  What  is  the  rise  in  temperature  of  the  field  coil  of  a  dynamo  which 
at  the  beginning  of  a  run  had  a  temperature  of  25°  C.  and  a  resistance  of 
290  ohms  if  its  final  resistance  is  346  ohms  ?    Use  a  =  0.0042. 

46.  The  P.D.  between  the  two  wires  of  the  circuit  shown  in  Figure  3 
is  2  volts  greater  at  a  than  at  b.    A  current  of  10  amperes  is  flowing  in  the 
circuit.    What  is  the  resistance  of  the  line  ? 


FIG.  3 

47.  If  it  is  desired  to  transmit  power  so  that  30  amperes  may  be  deliv- 
ered at  the  end  of  a  line  at  a  P.D.  of  110  volts,  what  must  be  the  resist- 
ance of  the  line  in  order  that  the  "line  drop"  shall  be  2  per  cent  of 
the  voltage  at  the  transmitting  end  ?     What  power   is   expended  in  the 
transmission  ? 

48.  A  group  of  incandescent  lamps  takes  15  amperes.    The  line  loss  is 
not  to  exceed  2  volts.    What  must  be  the  size  of  the  copper  wire  to  be  used 
if  the  lamps  are  2000  feet  from  the  transmitting  end  of  the  line  ?    (Given 
No.  5  wire,  182  mils  diameter,  and  No.  6  wire,  162  mils  diameter.)    How 
many  watts  are  lost  in  the  line  ? 


PROBLEMS 


359 


49.  No.  20  German  silver  wire  is  to  be  used  in  constructing  a  coil  of 
resistance  5  ohms  at  20° C.    If  the  diameter  of  the  wire  is  31.96  mils  and 
the  specific  resistance  of  the  specimen  is  0.00002076  per  centimeter  cube  at 
0°  C.,  how  many  centimeters  of  the  wire  will  be  necessary  ?    (a  =  0.0004.) 

50.  The  average  temperature  coefficient  of  platinum  is  0.00366.    What 
is  the  temperature  of  a  furnace  in  which  the  coil  of  a  platinum  thermom- 
eter has  a  resistance  of  1020  ohms?    The  resistance  at  0°C.  is  300  ohms. 

51.  The  resistance  of  the  platinum  thermometer  coil  of  Problem  50 
when  placed  in  a  bath  of  liquid  air  is  98- ohms.    What  is  the  temperature 
of  the  bath  ? 

52.  What  would  be  the  resistance  of  the  coil  at  the  limiting  tempera- 
ture of  the  absolute  ;:ero  ? 

53.  A  standard  resistance  of  0.0001  ohm  is  connected  in  a  circuit,  and 
a  millivoltmeter  connected   to   the   terminals   of   the   standard   resistance 
indicates  a  P.D.    of   0.137   volt.    Neglecting  the  resistance  of  the  milli- 
voltmeter, what  is  the  current  flowing  in  the  circuit? 

54.  A  universal  galvanometer  shunt  has  the  form  shown  in  Figure  4. 
The  conductor  k  attached  to  the  galvanometer  G  may  be  connected  to 
the  points  b,  c,  or  d. 

What  must  be  the 
ratio  of  .the  resist- 
ances ab,bc,  and  cd  in 
order  that  the  ratio 

—•-.— — -^ 

of  the  deflections  of  1  h 

the  galvanometer  for 
connections  at  6,  c, 
and  d  respectively 
shall  be  as  1:10: 100? 
(Consider  that  the 
resistance  ad  is  small 
compared  to  the  gal- 
vanometer resistance 
G.) 

55.  A  galvanom- 
eter having  a  resist- 
ance of  225  ohms  is  shunted  by  a  resistance  of  25  ohms.     Compare  the 
values  of  the   current  through   the  galvanometer  with  and  without  the 
shunt. 

56.  A  milliamrneter  reading  to  0.100  ampere  has  a  resistance  of  5  ohms. 
It  is  desired  to  use  this  instrument  to  measure  current  of  from  0  to  10  am- 
peres.   What  must  be  the  resistance  of  a  shunt  such  that  the  instrument  may 
be  read  directly  in  amperes  by  applying  a  multiplier  of  100  to  its  reading? 


FIG. 


360  ELECTRICITY,  SOUND,  AND  LIGHT 

57.  Two  voltmeters  of  resistance  20,000  and  16,000  ohms  respectively 
are  connected  in  series  across  220  volt  mains.    What  current  flows  through 
the  system,  and  what  is  the  reading  of  each  voltmeter? 

58.  If  the  voltmeters  of  Problem  57  are  connected  in  parallel  across 
220  volt  mains,  what  current  flows  through  each,  and  what  are  the  readings? 

59.  A  millivoltmeter   has  a  resistance  of   20  ohms.    What  resistance 
must  be  connected  in  series  with  it  in  order  that  it  may  be  used  to  read 
volts  directly  by  multiplying  its  readings  by  100? 


CHAPTERS  VIII  AND   IX 

60.  A  condenser  is  made  of  two  flat  metal  plates  separated  by  air.    If 
the  area  of  the  plates  is  500  sq.  cm.,  and  the  average  distance  between 
them  is  0.01  cm.,  what  is  the  capacity  of  the  condenser  in  electrostatic 
and  in  electro-magnetic  units  ? 

61.  If  a  plate  condenser  similar  to  that  described  in  Problem  -60  has  a 
capacity  of  0.002    microfarad,  and    another    condenser  with  glass  for  a 
dielectric  has  twice  the  area  of  plates  and  four  times  the  distance  of  sep- 
aration, and  has  a  capacity  of  0.0046  microfarad,  what  is  the  dielectric 
constant  of  the  glass  used  ? 

62.  What    quantity    of    electricity    would    charge    each   condenser   to 
110  volts? 

63.  What  is  the  ratio  of  the  resistance  of  a  conductor  as  measured  in 
the  E.S.  and  the  E.M.  systems  respectively? 

64.  Show  from  a  consideration  of  the  equation  C  =  — -  that  when  there 

is  a  given  charge  upon  the  plates  of  a  condenser,  the  value  of  the  field  be- 
tween the  plates  is  inversely  proportional  to  the  dielectric  constant  of  the 
medium  which  fills  the  space  between  the  plates. 

65.  The  charge  carried  by  the  hydrogen  ion  in  electrolysis  is  about 
4  x  10~10   absolute    electrostatic   units.     Express    this   charge   in    electro- 
magnetic units. 

66.  An  insulated  gold-leaf  electroscope  shows  a  charge  which  corresponds 
to  a  potential  of  450  volts.     When  an  insulated  sphere,  the  capacity  of 
which  is  15  electrostatic  units,  is  connected  with  the  gold  leaf,  the  poten- 
tial falls  to  the  point  corresponding  to  350  volts.    Find  the  capacity  of  the 
gold-leaf  electroscope. 

67.  When  the  terminals  of  a  dynamo  are  applied  directly  to  the  termi- 
nals of  an  electrostatic  voltmeter,  the  P.D.  indicated  is  500  volts.    An  oil 
condenser  consisting  of  plates  25  sq.  cm.  in  area  and  1  mm.  apart  is  placed 
in  series  with  the  electroscope.    When  the  terminals  of  the  same  dynamo 


PKOBLEMS  361 

are  applied  to  the  extremities  of  this  condenser  combination,  the  voltmeter 
indicates  50  volts.  If  the  dielectric  constant  of  the  oil  is  4,  what  is  the 
capacity  of  the  electrostatic  voltmeter  in  microfarads?  in  electrostatic 
units  ? 

68.  If  a  i  microfarad  condenser,  charged  by  means  of  a  2-volt  cell,  pro- 
duces a  throw  of  10  cm.  in  a  ballistic  galvanometer,  what  number  of 
microcoulombs  passed  through  the  galvanometer  when  it  was  thrown 
6  cm.  by  the  earth  inductor? 


CHAPTERS  X  AND  XI 

69.  A  battery  of  E.M.F.  1.07  volts  and  internal  resistance  1.8  ohms  is 
connected  to  a  coil  of  resistance  6  ohms.    What  is  the  P.D.  at  the  battery 
terminals,  and  what  current  flows  in  the  circuit? 

70.  A  tangent  galvanometer  of  resistance  1.5  ohms  is  deflected  54°  when 
connected  to  a  battery.    The  deflection  due  to  the  same  battery  is  only  42° 
when  an  additional  resistance  of  2  ohms  is  connected  in  the  circuit.    What 
is  the  resistance  of  the  battery  ? 

71.  A  galvanometer  of  20  ohms  resistance  gives  a  deflection  of  5°  when 
connected  to  a  battery  of  E.M.F.  1.08  volts  and  2  ohms  internal  resist- 
ance.   When  connected  to  a  second  battery  it  gives  a  deflection  of  8°. 
When  shunted  by  a  wire  of  20  ohms  resistance  and  connected  to  this 
second  battery  it  gives  a  deflection  of  7.5.    Find  the  E.M.F.  and  internal 
resistance  of  the  second  battery. 

72.  Six  Daniell  cells,  each  having  an  E.M.F.  of   1.08  volts  and  an 
internal  resistance  of  2  ohms,  are  to  be  connected  to  a  circuit  of  6  ohms 
resistance  by  wires  of  negligible  resistance.    What  is  the  current  through 
the  external  circuit  (a)  when  all  the  cells  are  in  series  ?   (b)  when  all  the 
cells  are  in  parallel  ?  (c)  when  they  are  arranged  in  three  groups  in  parallel, 
each  group  containing  two  cells  in  series  ?  (d)  when  arranged  in  two  groups 
in  parallel,  each  containing  three  cells  in  series?    What  is  the  P.D.  at  the 
terminals  of  the  battery  thus  formed  for  each  of  the  above  arrangements  ? 

73.  If  battery  cells  of  E.M.F.  2.16  volts  and  internal  resistance  1  ohm 
each  are  connected  as  in  Problem  72,  for  which  arrangement  is  the  current 
in  the  external  circuit  a  maximum,  and  what  is  its  value? 

74.  A  15  per  cent  solution  of  copper  sulphate  has  a  conductivity  of 
39  x  10~12  in  C.G.S.  units.     Calculate  the  current  due  to  a  P.D.  of  2  volts 
across  the  terminals  of  an  electrolytic  cell  of  this  solution  if  the  electrodes 
are  flat  plates  20  x  30  cm.  and  3  cm.  apart. 

75.  The  resistance  of  an  electrolytic  cell  similar  to  that  of  Problem  74 
is  1.94  ohms  at  18°  C.  and  0.80  at  100°  C.    Find  the  per  cent  increase 
in  conductivity  per  degree. 


362  ELECTRICITY,  SOUND,  AND  LIGHT 

76.  Three  Daniell  cells  of  E.M.F.  1.1  volts  and  internal  resistance  1.4 
ohms  each  are  connected  in  series  with  a  storage  battery  of   unknown 
internal  resistance  by  leads  of  0.3  ohm.    The   current  observed   is   1.17 
amperes.    The  storage  battery  terminals  are  then  reversed  and  a  current 
of  0.26  ampere  is  observed  to  now.    What  is  the  E.M.F.  of  the  storage 
battery,  and  what  P.D.  would  a  voltmeter  across  its  terminals  indicate  for 
each  connection  ? 

CHAPTERS  XII  AND  XIII 

77.  The  field  intensity  in  the  gap  space  between  the  poles  and  the 
armature   of   a  dynamo   is   5000  units.     Wires   25.5  cm.   in   length   and 
11  cm.  from  the  axis  of  the  armature  are  imbedded  in  the  iron  of  the 
armature  core.    The  armature  makes  1400  revolutions  per  minute.  'Find 
the  E.M.F.  in  volts  induced  in  each  armature  conductor  as  it  passes  under 
a  pole  face. 

78.  A  copper  disk  20  cm.  in  diameter  rotates  30  times  per  second  in  a 
uniform  field  of  intensity  700  units.    What  E.M.F.  is  induced  between  the 
center  and  the  circumference  of  the  disk  ? 

79.  A  coil  of  wire  of  30  turns  and  average  area  15  sq.  cm.,  carrying  a 
current  of  2  amperes,  is  suspended  from  the  beam  of  a  sensitive  balance 
so  that  the  plane  of  the  coil  is  horizontal.    If  the  intensity  of  the  earth's 
magnetic  field  is  0.57  and  the  angle  of  dip  is  63°,  what  is  the  difference 
in  weight  of  the  coil  produced  by  reversing  the  direction  of  the  current 
passing  through  it  ? 

80.  A  horizontal  wire  90  cm.  long  lies  in  a  northeast  direction.    How 
many  lines  of  force  does  it  cut  if  moved  vertically  upward  for  200  cm.? 
The  value  of  H  is  0.26  and  the  value  of  V  is  0.51. 


CHAPTERS  XIV  AND  XV 

81.  The  core  of  an  induction  coil  carries  100,000  lines  of   magnetic 
flux  when  a  given   current  flows  in  the  primary  of   the  coil.    Allowing 
0.004  sec.  for  the  flux  to  decrease  to  10,000  lines  after  the  primary  current 
has  been  interrupted,  how  many  turns  of  wire  are  necessary  on  the  second- 
ary of  the  coil  in  order  that  an  average  E.M.F.  of  20,000  volts  may  be 
induced  ? 

82.  A  coil  of  inductance  0.025  henry  and  1600  turns  of  wire  has  how 
many  lines  of  force  per  average  turn  for  a  current  of  35  amperes  ? 

83.  An  iron  ring  of  2.5  sq.cm.  cross  section  and  20  cm.  circumference 
has  a  primary  of  80  turns  and  a  secondary  of  4600  turns.     If  the  permea- 
bility of  the  iron  is  1600  for  a  current  of  1  ampere,  find  the  E.M.F.  induced 
in  the  secondary  by  reversing  the  primary  current  in  0.08  sec. 


PROBLEMS  363 

84.  An  iron  ring  of  0.75  cm.  radius  has  a  test  coil  wound  on  it  of  50 
turns  of  wire,  which  is  connected  to  a  ballistic  galvanometer.    The  ballistic 
galvanometer  circuit  has  a  resistance  of  300  ohms,  and  the  galvanometer 
throws  9  scale  divisions  when  0.000072  coulomb  passes  through  it.    What 
throw  is  caused  when  the  induction  in  the  iron  ring  changes  suddenly 
from  2400  lines  in  one  direction  to  3600  in  the  opposite  direction? 

CHAPTERS  XVII  AND  XVIII 

85.  Amagat  found  that  a  volume  of  a  cubic  centimeter  of  alcohol  at 
14°  C.  was  decreased  by  .000101  cc.  for  each  atmosphere  increase  in  the 
pressure  to  which  it  was  subjected.    If  the  density  of  alcohol  is  0.79,  what 
is  the  velocity  of  sound  in  this  medium? 

86.  Assuming  that  the  aural  impression  of  a  sound  persists  for  0.1  sec., 
calculate  the  distance  of  a  person  speaking  4  syllables  per  second  from  a 
reflecting  surface  in  order  that  the  echo  may  be  distinct.    Temperature  20°  C . 

87.  Find  the  change  in  pitch  observed  by  a  person  standing  on  a  rail- 
way platform  from  which  a  locomotive  with  whistle  blowing  is  receding 
at  the  uniform  rate  of  30  miles  per  hour. 

88.  Notes  of  225  and  336  vibrations  per  second  are  sounded  simulta- 
neously.   If  the  even  overtones  only  are  present  in  the  first  note,  and  both 
odd  and  even  overtones  in  the  second  note,  how  many  beats  per  second 
will  occur,  and  to  what  overtones  will  they  be  due  ? 

89.  Two  musicians  stationed  some  distance  apart  are  playing  slightly 
out  of  tune,  so  that  4  beats  per  second  are  noticeable.    How' fast  must  a 
person  travel  from  one  toward  the  other  in  order  that  no  beats  are  notice- 
able?   Calculate  for  notes  of  256  and  384  vibrations  per  second  at  a  tem- 
perature of  0°  C. 

90.  If  the  velocity  of   a  compressional  wave  in  a  gas  is  320  m.  per 
second  at  20°  C.,  what  will  be  the  velocity  at  50°  and  twice  the  pressure? 

91.  How  much  must  an  organ  pipe  be  heated  from  0°C.  in  order  that 
the  note  may  be  changed  by  a  semitone  ? 

92.  A  whistle  blown  normally  with  air  is  blown  with  hydrogen  of  den- 
sity 0.0692  as  compared  to  air  at  the   same  temperature   and   pressure. 
What  change  is  thus  produced,  and  what  is  the  interval  between  the  notes  ? 

CHAPTERS  XIX  AND  XX 

93.  A  brass  rod  2  m.   long  stroked  longitudinally  is  in  tune  with  a 
25-cm.  length  of  a  given  sonometer  wire.     A  steel  rod  3  m.  long  stroked 
longitudinally  is  in  tune  with  a  26-cm.  length  of  the  same  sonometer  wire. 
Find  the  relative  velocities  of  sound  in  steel  and  brass. 


364  ELECTKICITY,  SOUND,  AND   LIGHT 

94.  The  e  string  of  a  violin  has  a  mass  of  0.125  g.  and  a  length  of 
33  cm.  What  is  the  tension  in  kilograms  weight  which  it  exerts  when 
tuned  to  640  vibrations?  If  the  g,  d,  and  a  strings  were  to  be  made  out  of 
the  same  material  and  stretched  by  the  same  force,  what  would  be  the 
ratio  of  their  diameters  to  that  of  the  e  string? 


CHAPTERS  XXI  AND  XXII 

95.  The  theoretical  limit  of  resolution  of  a  lens  with  a  circular  aperture 
is  1.22  times   the   wave  length  of  light  divided  by  the  diameter  of  the 
aperture.     Mizar,  the  larger  of  the  two  stars  at  the  bend  of  the  handle  of 
the  Great  Dipper,  is  a  double  star.    Its  two  components  are  separated  by 
14.5"  of  arc.     What  is  the  smallest  aperture  of  telescope  that  can  resolve 
this  doublet? 

96.  If  the  distance  between  the  images  of  the  third  order  produced  by 
a  transmission  grating  placed  3  in.  from  a  sodium  flame  is  40cm.,  what 
is  the  number  of  lines  per  centimeter  in  the  grating? 

97.  The  two  D  lines  (sodium)  are  separated  by  an  angle  of  53"  of  arc 
in  the  first-order  spectrum  of  a  plane  diffraction  grating.    What  is  the 
grating  space  ? 

CHAPTERS  XXIII  AND  XXIV 

98.  If  the  absolute  index  of  refraction  of  glass  is  1.55  and  of  water 
1.33,  what  is  the  minimum  deviation  produced  in  a  beam  of  light  by  a 
prism  of  glass  immersed  in  water?    The  angle  of  the  prism  is  60°. 

99.  The  deviation  produced  by  a  prism  with  a  small  angle  is  commonly 
written  D  =  (/JL  —  1)^4.    Justify  this  equation. 

100.  When  a  layer  of  liquid  5  cm.  deep  is  placed  over  a  dot  on  a  glass 
plate,  the  position  of  the  dot  as  found  by  changing  the  focus  of  a  micro- 
scope is  1.45  cm.  above  the  plate.    What  is  the  index  of  refraction  of  the 
liquid? 

101.  From  the  law  "angle  of   incidence   equals  angle   of    reflection" 
deduce  the  fact  that  a  rotating  mirror  turns  through  one  half  the  angle 
through  which  the  reflected  ray  is  rotated. 

102.  What  is  the  focal  length  of  a  lens  which  has  conjugate  focal  lengths 
of  25  cm.  and  35  cm.? 

103.  What  is  the  length  of  the  real  image  of  a  line  4  cm.  long  and 
distant  40  cm.  from  a  lens  of  focal  length  25  cm.? 

104.  If   a  concave  mirror  has  a  curvature  of  0.05,  what  is  its  focal 
length?    How  far  from  the  mirror  will  be  formed  the  real  image  of  an 
object  distant  35  cm.  from  the  mirror?    What  will  be  its  size  relative  to 
that  of  the  object  ? 


PROBLEMS 


365 


105.  The  lens  L  forms  at  p'  a  real  image  of  p  (Fig.  5).     When  L  is 
moved  to  L'  it  again  forms  a  real  image  of /rat//.     If  LL'  is  60  cm.  and 

pp'  is  110  cm.,  what  is  the  focal 
P  I\L  IV  PI      length  of  lens  used ? 


FIG. 


106.  Parallel  rays  from  the 
sun  passing  through  the  open- 
ings a  and  b  (Fig.  6)  in  a  screen 

placed  before  a  concave  lens  L  illuminate  a  second  screen  at  c  and  d.    The 

distance  of  this  second  screen  from  the  lens  is  15  cm.    The  distances  ab 

and  cd  are  3  cm.  and  7.5  cm.  respectively. 

Find  the  focal  length  of  the  lens. 

107.  A  combination  of  a  doable  concave 
lens  and  a  more  powerful   convex   lens  of 
focal  length  15  cm.  has  a  focal  length  of 
80  cm.    Find  the  focal  length  of  the  con- 
cave lens. 

108.  An  object  A  is  placed  50  cm.  from 

a  concave  lens  L  of  unknown  focal  length  YIG.  0 

(Fig.  7).    A  concave  mirror  M  placed  upon 

the  opposite  side  of  the  lens  forms  a  real  image  of  the  object  at  B,  at  a 

distance  of  45  cm.  from  the  mirror.    The  focal  length  of  the  mirror  is 

30  cm.,  and  the  distance  from  the  mirror  to  the  lens  is  65  cm.     Find  the 

focal  length  of  the  lens. 


FIG. 


FIG.  8 


109.  Figure  8  shows  the  relative  position  of  an  object  A,  nearer  to  the 
lens  L  than  its  principal  focus  F,  and  its  virtual  image  B.  If  A  is  one 
third  the  focal  length  from  F,  what  are  the  relative  sizes  of  A  and  B? 


CHAPTERS  XXV  AND  XXVI 

110.  At  what  distance  from  a  photometer  must  a  Hefner  lamp  be  placed 
in  order  that  it  may  produce  the  same  intensity  of  illumination  as  &  stand- 
ard candle  distant  85  cm.  from  the  photometer? 

111.  Neglecting  the  motion  of  the  earth,  find  an  expression  for  the  ap- 
parent wave  length  X'  of  light  of  wave  length  A  from  a  distant  star  moving 
toward  the  earth  with  a  velocity  of  s.    Hence  write  an  expression  for  the 


366  ELECTRICITY,  SOUND,  AND  LIGHT 

velocity  of  the  star  in  the  line  of  sight.  From  this  equation  find  the 
velocity  in  the  line  of  sight  of  a  mass  of  hydrogen  in  the  neighborhood  of 
a  sun  spot  that  gave  in  the  spectrometer  an  apparent  wave  length  of 
6566  x  10~8  instead  of  6563  x  10~8,  which  corresponded  for  a  stationary 
source  to  the  dark  band  observed  in  the  solar  spectrum.  (See  Young, 
"  Manual  of  Astronomy,"  pp.  231-233.) 

112.  If   to  an   observer  on  the  earth,   who  is  by  the  earth's  motion 
moving  through  space  at  the  rate  of  18.5  miles  per  second,  light  from  a 
distant  star  seems  to  come  in  a  direction  making  an  angle  of  20.5"  with 
the  true  direction  of  the  star,  find  the  velocity  of  light  in  miles  per  hour. 
Sin  20. 5"  =  0.005842,   cos  20. 5"  =  0.999987.    (See   Young,    "Manual   of 
Astronomy,"  p.  150.) 

CHAPTERS  XXVII  AND  XXVIII 

113.  At  what  angle  must  a  beam  of  light  be  incident  upon  a  smooth 
water  surface  in  order  that  the  reflected  beam  shall  be  plane  polarized  ? 

114.  How  thick  should  be  a  piece  of  quartz  with  its  parallel  faces  cut 
perpendicular  in  order  that  plane  polarized  light  incident  upon  it  shall,  after 
transmission,  have  the  same  plane  of  vibration  as  before  ? 

115.  The  index  of  refraction  of  the  ordinary  ray  in  Iceland  spar  is 
1.6543.    The  index  of  Canada  balsam  is  1.536.    At  what  angle  must  the 
ordinary  ray  in  a  Nicol  prism  meet  the  interface  in  order  that  it  may  just 
suffer  total  reflection? 

116.  If  an  a  particle  projected  from  uranium  produces  on  the  average 
100,000  positive  ions,    how  many  a  particles  are  required  to  produce  the 
ionization  current  given  in  the  example  at  the  end  of  Experiment  28? 


TABLES 


Table  1 
SATURATED  WATER  VAPOR 

Showing  pressure  P  (in  mm.  of  mercury)  and  density  D  of  aqueous  vapor  saturated 
at  temperature  t;  or  showing  boiling  point  t  of  water  and  density  D  of  steam  cor- 
responding to  an  outside  pressure  P. 


t 

P 

D 

t 

P 

D 

t 

p 

D 

—10 

2.2 

2.3X10-6 

30 

31.5 

30.1X10-6 

88.5 

496.2 

—  9 

2.3 

2.5  " 

35 

41.8 

39.3  " 

89 

505.8 

—  8 

2.5 

2.7  " 

40 

54.9 

50.9  " 

89.5 

515.5 

—  7 

2.7 

2.9  " 

45 

71:4 

65.3  " 

90 

525.4 

428.4X10-6 

—  6 

2.9 

3.2  " 

50 

92.0 

83.0  " 

90.5 

535.5 

—  5 

3.2 

3.4  " 

55 

117.5 

104.6  " 

91 

545.7 

-  4 

3.4 

3.7  " 

60  ' 

148.8 

130.7  " 

91.5 

556.1 

—  3 

3.7 

4.0  " 

65 

187.0 

162.1  " 

92 

566.7 

—  2 

3.9 

4.2  " 

70 

233.1 

199.5  " 

92.5 

577.4 

—  1 

4.2 

4.5  " 

71 

243.6 

93 

588.3 

0 

4.6 

4.9  " 

72 

254.3 

93.5 

599.6 

1 

4.9 

5.2  " 

73 

265.4 

94 

610.6 

2 

5.3 

5.6  " 

74 

276.9 

94.5 

622.0 

3 

5.7 

6.0  " 

75 

288.8 

243.7  " 

95 

633.6 

511.1  " 

4 

6.1 

6.4  " 

75.5 

294.9 

95.5 

645.4 

5 

6.5 

6.8  " 

76 

301.1 

96 

657.4 

6 

7.0 

7.3  " 

76.5 

307.4 

96.5 

669.5 

7 

7.5 

7.7  " 

77 

313.8 

97 

681.8 

8 

8.0 

8.2  " 

77.5 

320.4 

97.5 

694.2 

9 

8.5 

8.7  " 

78 

327.1 

98 

707.1 

10 

9.1 

9.3  " 

78.5 

333.8 

98.2 

712.3 

11 

9.8 

10.0  " 

79 

340.7 

98.4 

717.4 

12 

10.4 

10.6  " 

79.5 

347.7 

98.6 

722.6 

13 

11.1 

11.2  " 

80 

354.9 

295.9  " 

98.8 

727.9 

14 

11.9 

12.0  " 

80.5 

362.1 

99 

733.2 

15 

12.7 

12.8  " 

81 

369.5 

99.2 

738.5 

16 

13.5 

13.5  " 

81.5 

377.0 

99.4 

743.8 

17 

14.4 

14.4  " 

82 

384.6 

99.6 

749.2 

48 

15.3 

15.2  " 

82.5 

392.4 

99.8 

754.7 

19 

16.3 

16.2  " 

83 

400.3 

100 

760.0 

606.2  " 

20 

17.4 

17.2  " 

83.5 

408.3 

100.2 

765.5 

21 

18.5 

18.2  " 

84 

416.5 

100.4 

771.0 

22 

19.6 

19.3  " 

84.5 

424.7 

100.6 

776.5 

23 

20.9 

20.4  " 

85 

433.2 

357.1  " 

100.8 

782.1 

24 

22.2 

21.6  " 

85.5 

441.7 

101 

787.7 

25 

23.5 

22.9  " 

86 

450.5 

102 

816.0 

26 

25.0 

24.2  " 

86.5 

459.3 

103 

845.3 

27 

26.5 

25.6  " 

87 

468.3 

105 

906.4 

715.4  " 

28 

28.1 

27.0  " 

87.5 

477.4 

107 

971.1 

29 

29.7 

28.5  " 

88 

486.8 

110 

1075.4 

840.1  " 

368 


ELECTBICITY,  SOUND,  AND  LIGHT 


Table  2 
DENSITY  OF  DRY  AIR  AT  TEMPERATURE  t  AND  PRESSURE  If  mm.  OF  MERCURY 


t 

£"=720 

730 

740 

750 

760 

770 

DIFFERENCE 
PER  mm. 

10° 

.001181 

.001198 

.001214 

.001231 

.001247 

.001263 

16 

11 

1177 

1194 

1210 

1226 

1243 

1259 

1    2 

12 

1173 

1189 

1206 

1222 

1238 

1255 

2     3 

13 

1169 

1185 

1202 

1218 

1234 

1250 

3     5 

14 

1165 

1181 

1197 

1214 

1230 

1246 

4     6 

15° 

.001161 

.001177 

.001193 

.001209 

.001225 

.001242 

5     8 

16 

1157 

1173 

1189 

1205 

1221 

1237 

6    10 

17 

1153 

1169 

1185 

1201 

1217 

1233 

7    11 

18 

1149 

1165 

1181 

1197 

1213 

1229 

8    13 

19 

1145 

1161 

1177 

1193 

1209 

1224 

9    14 

20° 

.001141 

.001157 

.001173 

.001189 

.001204 

.001220 

15 

21 

1137 

1153 

1169 

1185 

1200 

1216 

1    2 

22 

1133 

1149 

1165 

1181 

1196 

1212 

2     3 

23 

1130 

1145 

1161 

1177 

1192 

1208 

3     4 

24 

1126 

1141 

1157 

1173 

1188 

1204 

4     6 

25° 

.001122 

.001138 

.001153 

.001169 

.001184 

.001200 

5     7 

26 

1118 

1134 

1149 

1165 

1180 

1196 

6     9 

27 

1114 

1130 

1145 

1161 

1176 

1192 

7    10 

28 

1110 

1126 

1142 

1157 

1172 

1188 

8    12 

29 

1107 

1122 

1138 

1153 

1169 

1184 

9    13 

30° 

.001103 

.001119 

.001134 

.001149 

.001165 

.001180 

Correction  for  Moisture  in  Above  Table 


Dew-point 

Subtract 

Dew-point 

Subtract 

Dew-point 

Subtract 

Dew-point 

Subtract 

-10° 
—  8 
—  6 

—  4 

—  2 

.000001 
.000002 
.000002 
.000002 
.000003 

0° 

+2 
+4 
+6 
-f8 

.000003 
.000003 
.000004 
.000004 
.000005 

+10° 
+  12 
+14 

+  16 

+18 

.000006 
.000006 
.000007 
.000008 
.000009 

+20° 
+22 
+24 
+26 
+28 

.000010 
.000012 
.000013 
.000015 
.000016 

Table  3 
DENSITY  OF  WATER 


Table  4 
DENSITY  OF  MERCURY 


TEMP.  C° 

DENSITY 

TEMP.  C° 

DENSITY 

TEMP.  C° 

DENSITY 

0° 

0.999884 

13° 

0.999443 

0° 

13.596 

1 

0.999941 

14 

0.999312 

10 

13.572 

2 

0.999982 

15 

0.999173 

12 

13.567 

3 

1.000004 

16 

0.999015 

14 

13.562 

3.95 

1.000000 

17 

0.998854 

16 

13.557 

4 

1.000013 

18 

0.998667 

18 

13.552 

5 

1.000003 

19 

0.998473 

20 

13.547 

6 

0.999983 

20 

0.998272 

22 

13.542 

7 

0.999946 

22 

0.997839 

24 

13.537 

8 

0.999899 

24 

0.997380 

26 

13.532 

9 

0.999837 

26 

0.996879 

28 

13.528 

10 

0.999760 

28 

0.996344 

30 

13.523 

11 

0-999668 

30 

0.995778 

32 

13.518 

12 

0.999562 

100 

0.958860 

34 

13.513 

TABLES 


369 


Table  5 

DENSITIES  AND  ELECTRICAL  CONDUCTIVITIES  AT  18C 
(FROM  KOHLRAUSCH) 


C.  OF  NORMAL  SOLUTIONS 


A  =  equivalent  weight  (O=  16.00)  or  concentration  in  gram  equivalents  per  liter. 

p  =  number  of  grams  of  substance  to  100  grams  of  the  solution. 

d  =  density  in  grams  per  cubic  centimeter  at  18°  C. 

<r  =  electrical  conductivity,  i.e.  reciprocal  of  specific  resistance  in  ohms  per  cubic 

centimeter 
Aa  =  relative  increase  in  <r  per  degree  in  neighborhood  of  18°  C. 


A 

P 

d 

10*0- 

Aa 

(7 

KOH     
KC1 

56.16 
74  60 

5.359 

7  139 

1.0479 
1  0449 

184.0 
98  3 

.0186 
0193 

KNOg    
NH4C1  
NaOH  
NaCl     
NaC2H3O2     .    . 

101.19 
53.52 
40.06 

58.50 
82  05 

9.544 
5.271 
3.844 
5.629 

7  897 

1.0602 
1.0153 
1.0420 
1.0392 
1  0400 

80.5 
97  0 
160.0 
74.3 

41  2 

.0200 
.0194 
.0197 
.0212 
0250 

-'Na2S04  
LiCl      

71.08 

42.48 

6.703 
4  157 

1.0604 
1  0226 

50.8 
63  4 

.0236 
0220 

iCaC!2     
i  ZnCl2 

55.45 
68  15 

5.313 
6  442 

1.0436 
1  0578 

67.8 
55  0 

.0207 
0220 

iZnSO4    
1  CuS04 

80.73 
79  83 

7.483 

7  408 

1.0789 
1  0776 

26.6 
25  8 

.0220 
0220 

AgNO8 

169  97 

14  91 

1  1400 

67  8 

Ot;>10 

HC1       
HN03    
1H2S04     
C2H4O2 

36.46 
63.05 
49.04 
60  03 

3.587 
6.  107 
4.758 
5  959 

1.0165 
1.0325 
1.0307 
1  0074 

300.0 
299.0 
197.0 

.0159 
.0150 
.0120 

Suffar 

342  2 

30  30 

1  1294 

Table  6 

VISCOSITIES  OF  AQUEOUS  SOLUTIONS  OF  VARYING  CONCENTRATIONS 

If  z  is  the  viscosity  of  a  dilute  solution  in  terms  of  the  viscosity  of  water,  and  x  the 
concentration  in  gram  molecules  per  liter,  then,  according  to  Arrhenius, 


A,  therefore,  represents  the  viscosity  of  a  normal  solution  divided  by  the  viscosity  of 
pure  water. 

The  values  in  the  following  table  refer  to  viscosities  at  25°  C. 


SUBSTANCE 

A 

SUBSTANCE 

A 

Alcohol  {C2H5OH} 
Cane  Sugar  {Ci2H22On} 
Copper  Nitrate  {Cu(NO3)2} 
Copper  Sulphate  {CuSO4} 
Ether  {(C2H5)20| 
Hydrochloric  Acid  {HC1} 
Nitric  Acid  |HNO3} 
Potassium  Nitrate  |KNO3} 

1.030 
1.046 
1.1729 
1.3533 
1.026 
1  0699 
1.0233 
0.9664 

Potassium  Sulphate  {K2SO4} 
Sodium  Acetate  {Na2C2H3()2} 
Sodium  Chloride  {NaCl} 
Sodium  Nitrate  {NaNO3} 
Sodium  Sulphate  *Na2S04} 
Zinc  Nitrate  {Zn(NO3)2} 
Zinc  Sulphate  {ZnS04| 

1.0982 
1.3998 
1.0986 
1.0522 
1.2253 
1.1666 
1.3613 

370 


ELECTRICITY,  SOUND,  AND  LIGHT 


Table  7 

ATOMIC  WEIGHTS  (0  =  16.00) 


(International  1904) 


Aluminum      .    .    . 

.    .    Al 

27.1 

Antimony  .... 

Sb 

120.2 

Argon     

.    .    A 

39.9 

Arsenic  

.    .    As 

75.0 

Barium  

.    .    Ba 

137.4 

Bismuth      .... 

.     .    Bi 

208.5 

Boron     

.    .    B 

11. 

Bromine     .... 

.    .    Br 

79.96. 

Cadmium   .... 

.    .    Cd 

112.4 

Caesium  

.    .    Cs 

132.9 

Calcium      .... 

Ca 

40.1 

Carbon  

.    .    C 

12.00 

Cerium  

Ce 

140.25 

Chlorine     .... 

.    .    Cl 

35.45 

Chromium      .    .    . 

Cr 

52.1 

Cobalt    

Co 

59.0 

Columbium    .    .    . 

.    .    Cb 

94. 

Copper  

.    .    Cu 

63.6 

Erbium  

.    .    Er 

166. 

Fluorine     .... 

.    .    F 

19. 

Gadolinium    .    .    . 

.     .    Gd 

156. 

Gallium      .... 

.    .    Ga 

70. 

Germanium    .    .    . 

.    .    Ge 

72.5 

Glucinum  .... 

.    .    Gl 

9.1 

Gold  ...... 

.    .    Au 

197.2 

Helium  

.    .    He 

4. 

Hydrogen  .... 

H 

1.008 

Indium  

.    In 

114. 

Iodine     

.    .    I 

126.85 

Iridium  

Ir 

193.0 

Iron    

.    .    Fe 

55.9 

Krypton      .... 

Kr 

81.8 

Lanthanum    .    .     . 

.    .    La 

138.9 

Lead  

Pb 

206.9 

Lithium     .    .    .    . 

Li 

7.03 

Magnesium     .    .     . 

.    .    Mg 

24.36 

Manganese      .    .    . 

.    .    Mn 

55.0 

Mercury     .... 

.    .    Hg 

200.0 

Molybdenum      .    . 

.    .    Mo 

96.0 

Neodymium   .....  Nd  143.6 

Neon Ne         20. 

Nickel     . Ni         58.7 

Nitrogen N          14.04 

Osmium Os  191. 

Oxygen 0  16.00 

Palladium Pd  106.5 

Phosphorus P  31.0 

Platinum Pt  194.8 

Potassium K          39.15 

Praseodymium  .    .    .    .  Pr  140.5 

Radium Rd  225. 

Rhodium Rh  103.0 

Rubidium Rb        85.4 

Ruthenium Ru  101.7 

Samarium Sm  150. 

Scandium So         44.1 

Selenium Se        79.2 

Silicon Si         28.4 

Silver Ag  107.93 

Sodium Na        23.05 

Strontium Sr         87.6 

Sulphur S  32.06 

Tantalum Ta  183. 

Tellurium  ......  Te  127.6 

Terbium Tb  160. 

Thallium Tl  204.1 

Thorium Th  232.5 

Thulium Tm  171. 

Tin Sn  119.0 

Titanium Ti         48.1 

Tungsten W  184.0 

Uranium U  238.5 

Vanadium V          51.2 

Xenon Xe  128. 

Ytterbium Yb  173.0 

Yttrium Yt         89.0 

Zinc Zn         65.4 

Zirconium .  .  Zr         90.6 


TABLES 


371 


Table  8 

ELECTRO-CHEMICAL  EQUIVALENTS 


SUBSTANCE 

VALENCY 

GRAMS  PER 
COULOMB 

SUBSTANCE 

VALENCY 

GRAMS  PER 
COULOMB 

Aluminum  . 

3 

.0009357 

Magnesium  . 

2 

.0001261 

Bromine  .    . 

1 

.0008276 

Manganese    . 

2 

.0002849* 

Cadmium     . 

2 

.000582* 

Mercury   .    . 

1 

.002075 

Chlorine  .    . 

1 

.0003672 

Mercury   .    . 

2 

.001037 

Cobalt      .    . 

2 

.0003056* 

Nickel  .    .    . 

2 

.0003042 

Copper     .    . 

1 

.0006588 

Nitrogen  .    . 

3 

.0000485* 

Copper     .    . 

2 

.0003294 

Oxygen     .    . 

2 

.0000829 

Fluorine 

1 

.0001968* 

Platinum  .    . 

4 

.000500* 

Gold    .    .    . 

3 

.000681 

Potassium     . 

1 

.0004054 

Hydrogen    . 

1 

.0000104 

Silver    .    .    . 

1 

.001118 

Iodine      .    . 

1 

.001314 

Sodium     .    . 

1 

.0002387 

Iron     .    .    . 

2 

.0002895 

Tin    .... 

2 

.0006163 

Iron     .    .    . 

3 

.0001930 

Tin    .... 

4 

.0003081 

Lead    .    .    . 

2 

.001072 

Zinc  .... 

2 

.0003387 

*  Starred  values  are  calculated. 

Table  9 

SPECIFIC  RESISTANCES  AND  TEMPERATURE  COEFFICIENTS 

Metals 


SUBSTANCE 

RESISTANCE  AT 
0°  C.  IN  OHMS  PER 
CENTIMETER  CUBE 

MEAN  TEMPERA- 
TURE COEFFICIENT 
0°-100°C. 

Aluminum    
Antimony 

2.906  x  10-  6 
35  42    x  10~  6 

.00435 

Bismuth    .    . 

130  9      x  10-  6 

Copper  (annealed) 

1  584  x  10-  6 

0042 

Copper  (hard  drawn)    
Iron  (annealed)    
Iron  (hard  drawn)    
Gold      .    . 

1.619  x  lO-6 
9.693  x  lO-6 
15.         x  10-  6 
2  088  x  10-  6 

.00625 
00377 

Mercury    
Nickel  
Platinum       

Silver 

94.34    x  10-  6 
12.35    x  10-6 
9.035  x  lO-6 
1  561  x  10~G 

.0009 
.00622 
.00367 
00400 

Tin 

10  5      x  10-  6 

.00440 

Zinc  . 

5  75    x  10-  6 

.00406 

German  Silver      
Platinum  Silver  (Pt  33%,  Ag  66%)  .    . 

Platinum  Iridium  (Pt  80%,  Ir  20%)  .    . 
Platinum  Rhodium  (Pt  90%,  Rh  10%)  . 
Manganin  (Cu  84%,  Mn  12%,  Ni  4%)     . 

20.89    x  10-° 
31.6      x  10-6 
30.9      x  10-6 
21.1      x  10-6 
46.7      x  lO-6     ; 

.00027 

.00082 
.00143 
.0000 

372 


ELECTRICITY,  SOUND,  AND  LIGHT 


Table  9  (continued) 
Electrolytes 

R  =  resistance  in  ohms  per  centimeter  cube;  «=per  cent  decrease  in  resistance  per 
degree  Centigrade  ;  solutions  given  in  per  cent  by  weight  of  salts  free  from  water  of  crys- 
tallization 


5 

fc 

10 

% 

1 

5% 

R 

a 

R 

a 

R 

« 

KC1 

14.5 

2 

7.34 

1.9 

4.95 

1.9 

NaCl                    .... 

14.9 

2.2 

8.27 

2.1 

6.10 

2.1 

NH4C1         

10.9 

2. 

5.62 

1.9 

3.86 

1.7 

CuSO4        

52.6 

2.2 

31.2 

2.2 

23.8 

2.3 

ZnS04    
MgS04   
AgN03   
KOH                             .    . 

52.6 
38.4 
38.5 
5.81 

2.2 
2.3 

2.2 
1.9 

31.3 
24.4 

20.8 
3.17 

2.2 
2.4 

2.2 
1.9 

23.8 
20.8 
14.7 
2  35 

2.2 
2.5 
2.2 
1  9 

HC1        

2.53 

1.58 

1.59 

1.56 

1  34 

1.55 

HNO8     
H2S04    

3.88 
4.78 

1.5 
1.21 

2.17 

2.55 

1.45 

1.28 

1.63 
1.84 

1.4 
1.36 

Table  10 


DIELECTRIC  CONSTANTS 


Air     

Alcohol                        .    . 

1.00 
26.5 
2     -3 
3-5 

.      2.2-2.7 
•2.     -8. 
.      1.9-2. 
.      2.1-2.2 

Tabl 

YOUNG'S 

Quartz 

4  5 

3.7 
4. 
2.3 

1.015 

Shellac    . 

27        — 

Ebonite      
Common  glass   .... 
India  rubber      .... 
Mica       

Sulphur  .    .    . 
Benzol 

.      2. 

22        — 

Turpentine  .    . 
Water      .    .    . 

.      2.2 
.    81 

Paraffin      
Petroleum  oil     .... 

Gases       .    .    . 
Vacuum  .    .    . 

.9998  - 
0.9985 

e  11 

MODULUS 

Approximate  Values  in  Dynes  per  Square  Centimeter 


Cast  iron      

Brass  wire 

Copper  wire 

Steel  wire 

Aluminum 

Nickel 

Silver 

Glass 

Oak  (cut  longitudinally) 
Pine  (cut  longitudinally) 


12     x  10H 

10     x  1011 

12      x  ion 

19-20      x  ion 

6.5  x  ion 

20      x  ion 

7.3  x  ion 

6.5  x  ion 

0.9  x  10n 

0.5-0.6  x  1011 


TABLES 


373 


(about) 


Aluminum  . 
Bismuth  .  . 
Brass  .  .  . 
Brick  .  .  . 

Copper       

Cork 

Diamond 

Glass  (common  crown) 
"     (flint)  .... 

Gold      

Ice  at  0°  C 

Iron  (cast)     .... 


Table  12 


DENSITIES 


Solids 


2.58 
9.80 
8.5 
2.1 
8.92 
0.24 
3.52 
2.6 

3.0-6.3 
19.3 
0.9167 
7.4 


Iron  (wrought) 
Lead.  .  .  . 
Nickel  .  .  . 
Oak  .... 
Pine  .... 
Platinum  .  . 
Quartz  .  .  . 
Silver  .  .  . 
Sugar  .  .  . 
Sulphur  .  . 
Tin  .... 
Zinc  . 


7.86 

11.3 

8.9 

0.8 

0.5 

21.50 

2.65 

10.53 

1.6 

2.07 

7.29 

7.15 


Alcohol  at  20°  C.  . 
Carbon  bisulphide 
Ethyl  ether  at  0°  C. 
Glycerin    .    .    . 
Turpentine    .    .    . 
Benzol  . 


Liquids 


0.789 

Gasoline  .    . 

79 

1.29 
0.735 
1.26 

0.87 

Mercury  
Sulphuric  acid     .    .    . 
Hydrochloric  acid  .    . 
Nitric  acid  .    .    . 

.    .    13.596 
.    .      1.85 
.    .    .      1.27 
1  56 

.88 

Olive  oil  . 

0.91 

Gases  at  0°  C.  76  cm.  of  Mercury  Pressure 


Acetylene  (C2Ho)     ....  0.001185 

Ammonia  (NH3) 0.000770 

Carbon  monoxide  (CO)    .     .  0.001252 

Carbon  dioxide  (C02)  .    .    .  0.001974 


Hydrogen  (H2)  . 
Marsh  gas  (CH4) 
Nitrogen  (N2) 
Oxygen  (O2)   .    . 


0.0000895 
0.000715 
0.001257 
0.001430 


Table  13 


AVERAGE  SPECIFIC   HEATS 


Alcohol  .... 
Aluminum  . 
Bismuth 

Brass  .... 
Copper  .... 
Carbon  bisulphide 
Ebonite  .... 
German  silver . 

Glass 

Gold 

Graphite      .     .     . 

Ice 

Iron  . 


at  40° 

0.648 

Lead  . 

QO  _  100° 

0.2185 

Magnesium 

0.0303 

Mercury  . 

0.094 

Nickel 

QO  _  IQQO 

0.095 

Paraffin  . 

40° 

0.2429 

Petroleum 

0.33 

Platinum 

0.0946 

Quartz     . 

0.20 

Silver  .     . 

0.0316 

Steel   .     . 

11° 

0.160 

Tin      .     . 

0.504 

Turpentine 

0°  -  100° 

0.1130 

Zinc    . 

25° 
10° 

21° 

20° 


100° 

0.0315 

0.251 

50° 

0.0333 

100° 

0.1128 

0.683 

58° 

0.511 

100° 

0.0323 

100° 

0.19 

100° 

0.0568 

0.118 

100° 

0.0559 

0.467 

0.0935 

374  ELECTRICITY,  SOUND,  AND  LIGHT 

Table  14 

WAVE  LENGTH  IN  MICRONS  (^u)  OR  THOUSANDTHS  MILLIMETERS 


FRAUN- 

HOFEK 

LINE 

SOURCE 

WAVE 
LENGTH 

IN  fJi 

FRAUN- 

HOFER 

LINE 

SOURCE 

WAVE 
LENGTH 

IN  /* 

Potassium 

.7699 

61 

Magnesium 

.5184 

Potassium 

.7665 

62 

Magnesium 

.5173 

A 

Sun 

.7604 

63 

Iron 

.5169 

a 

Sun 

.7185 

&4 

Magnesium,  iron 

.5168 

B 

Oxygen 

.6870 

Cadmium 

.5086 

Lithium 

.6708 

Barium 

.4934 

C 

Hydrogen 

.6563 

Calcium 

.4878 

Strontium 

.6550 

F 

Hydrogen 

.4861 

Calcium 

.6499 

Strontium 

.4607 

Cadmium 

.6438 

Barium 

.4554 

Strontium 

.6408 

Mercury 

.4358 

a 

Oxygen 

.6278 

f 

Hydrogen 

.4340 

Mercury 

.6152 

G' 

Iron 

.4326 

Lithium 

.6104 

G 

Iron,  calcium 

.4308 

Di 

Sodium 

.5896 

Calcium 

.4227 

Dz 

Sodium 

.5890 

h 

Hydrogen 

.4102 

Ds 

Helium 

.5876 

Mercur^ 

.4078 

Mercury 

.5790 

Mercury 

.4047 

Mercury 

.5769 

Potassium,  iron 

.4046 

Mercury 

.5461 

H 

Hydrogen,  calcium 

.3968 

Cadmium 

.5379 

Cadmium 

.5338 

TT 

Iron,  calcium 

.5270 

Jjj 

Iron 

.5270 

Table  15 

REFRACTIVE  INDICES  FOR  DIFFERENT  COLORS 


RED(C) 

YELLOW(Z>) 

BLUE  (F) 

Water  .... 

1  3317 

1  3335 

1  3377 

Alcohol  .     . 

1  3606 

1  3624 

1  3667 

Carbon  bisulphide  ... 

1  6198 

1  6293 

1  6541 

/light      
Crown  glass  H  . 
I  heavy    .    . 

1.5127 
1.6126 

1.5153 
1  6152 

1.5214 
1  6213 

f  lisrht 

1  6038 

1  6085 

1  6200 

Flint  glass  \     * 
1  heavy  

1  7434 

1  7515 

1  7723 

r  ordinary  ray    . 
Iceland  spar  4 
i.  extraordinary  ray    . 

Quartz.f0rdinallra7     
i.  extraordinary  ray     .    .    . 

1.6545 
1.4846 
1.5418 
1.5509 

1.6585 
1.4864 
1.5442 
1.5533 

1.6679 
1.4908 
1.5496 
1.5589 

TABLES 


375 


Table  16 

REDUCTION  OP  BAROMETRIC  HEIGHT  TO  0"  C. 

(The  table  corrections  represent  the  number  of  millimeters  to  be  subtracted  from 
the  observed  height  h.  They  are  obtained  from  the  formula  (.000181  — .000019)^,  the 
first  number  being  the  cubical  expansion  coefficient  of  mercury,  the  second  the  linear 
coefficient  of  brass.) 


t 

OBSERVED  HEIGHT  IN  mm. 

680 

690 

700 

710 

720 

730 

740 

750 

760 

770 

mm. 

mm. 

mm. 

mm. 

mm. 

mm. 

mm. 

mm. 

mm. 

mm. 

10° 

1.10 

1.12 

1.13 

1.15 

1.17 

1.18 

1.20 

1.22 

1.23 

1.25 

11 

1.21 

1.23 

1.25 

1.27 

1.28 

1.30 

1.32 

1.34 

1.35 

1.37 

12 

1.32 

1.34 

1.36 

1.38 

1.40 

1.42 

1.44 

1.46 

1.48 

1.50 

13 

1.43 

1.45 

1.47 

1.50 

1.52 

1.54 

1.56 

1.58 

1.60 

1.62 

14 

1.54 

1.56 

1.59 

1.61 

1.63 

1.66 

1.68 

1.70 

1.72 

1.75 

15 

1.65 

1.68 

1.70 

1.73 

1.75 

1.77 

1.80 

1.82 

1.85 

1.87 

16 

1.76 

1.79 

1.81 

1.84 

1.87 

1.89 

1.92 

1.94 

1.97 

2.00 

17 

1.87 

1.90 

1.93 

1.96 

1.98 

2.01 

2.04 

2.07 

2.09 

2.12 

18 

1.98 

2.01 

2.04 

2.07 

2.10 

2.13 

2.16 

2.19 

2.22 

2.25 

19 

2.09 

2.12 

2.15 

2.19 

2.22 

2.25 

2.28 

2.31 

2.34 

2.37 

20 

2.20 

2.24 

2.27 

2.30 

2.33 

2.37 

2.40 

2.43 

2.46 

2.49 

21 

2.31 

2.35 

2.38 

2  42 

2.45 

2.48 

2.52 

2.55 

2.59 

2.62 

22 

2.42 

2.46 

2.49 

2'.  53 

2.57 

2.60 

2.64 

2.67 

2.71 

2.74 

23 

2.53 

2.57 

2.61 

2.65 

2.68 

2.72 

2.76 

2.79 

2.83 

2.87 

24 

2.64 

2.68 

2.72 

2.76 

2.80 

2.84 

2.88 

2.92 

2.95 

2.99 

25 

2.75 

2.79 

2.84 

2.88 

2.92 

2.96 

3.00 

3.04 

3.08 

3.12 

Table  17 

USEFUL  CONSTANTS  AND  RELATIONS 
TT  =  3.1416.     7T2  =  9.8696,  -  =  0.31831.     logarithm  TT  =  .49715. 

7T 


Naperianbasee  =  2.7183.     Iog10e  =  .  43429. 


=  2.30261og10#. 


1  inch  =  25.4  millimeters.  1  meter  =  39.37  inches.  1  mile  =  1.609  kilometers. 
1  kilogram  =  2.2  pounds.  1  pound  =  453.59  grams.  1  ounce  =  28.35  grams. 
1  grain  =  64.8  milligrams.  Mechanical  equivalent  of  lcalorie(15°)  =4.19  x  107  ergs. 

1  horse  power  =  746  watts  =  33000  foot  pounds  per  minute. 

1  watt  =  1  joule  per  second.     1  joule  =  107  ergs  =  0.7373  foot  pounds. 

1  radian  =  57.3  degrees.     180°  =  TT  radians. 

Half  diameter  of  the  earth,  equatorial,  6378.2  kilometers;  polar,  6356.5  kilo- 
meters; mean,  6367.4  kilometers. 

Mean  density  of  the  earth,  5.5270. 


376  ELECTRICITY,  SOUND,  AND  LIGHT 

Table  18 

RESISTANCES  OF  COPPER  AND  OF  GERMAN  SILVER  WIRE 

Brown  and  Sharp  Gauge 


PURE  COPPER 

18%  GERMAN  SILVER 

Number 

Diameter  in  Mils 
data  in-) 

Ohms  per  1000  ft. 

Ohms  per  1000  ft. 

000 

409.64 

.064 

00 

•    354.80 

.081 

o 

324.95 

.102 

\ 

289.30 

.129 

2 

257.63 

.163 

ooq  40 

4 

±.~-'  .  t  — 
204.31 

.259 

, 

mQ4 

Q0« 

.  «7:± 

162.02 

.  O£\J 

411 

7 

144.28 

•  *±  _L  1. 
FL1Q 

i 

128.49 

•  O  L  U 

.654 

9 

114^43 

]824 

11.832 

10 

101.89 

1.040 

18.720 

11 

90.74 

1.311 

23.598 

12 

80.81 

1.653 

29.754 

13 

71.96 

2.084 

37.512 

14 

64.08 

2.628 

47.304 

15 

57.07 

3.314 

59.652 

16 

50.82 

4.179 

75.222 

17 

45.26 

5.269 

94.842 

18 

40.30 

6.645 

119.610 

19 

35.89 

8.617 

155.106 

20 

31.96 

10.566 

190.188 

21 

28.46 

13.323 

239.814 

22 

25.35 

16.799 

302.382 

23 

22.57 

21.185 

381.330 

24 

20.10 

26.713 

480.834 

25 

17.90 

33.684 

606.312 

26 

15.94 

42.477 

764.586 

27 

14.20 

53.563 

964.134 

28 

12.64 

67.542 

1215.756 

29 

11.26 

85.170 

1533.060 

30 

10.03 

107.391 

1933.038 

31 

8.93 

135.402 

2437.236 

32 

7.95 

170.765 

3073.770 

33 

7.08 

215.312 

3875.616 

34 

6.30 

271.583 

4888.494 

35 

5.61 

342.443 

6163.974 

36 

5.00 

431.712 

7770.816 

37 

4.45 

544.287 

9797.166 

38 

3.97 

686.511 

12357.198 

39 

3.53 

865.046 

15570.828 

40 

3:14 

1091.865 

19653.570 

TABLES 


377 


Table  19 

RESUME  OF  DEFINING  EQUATIONS  AND  UNITS 


MAGNITUDE 

DEFINING  EQUA- 
TION 

SYMBOL  FOR 
IN  C.G.S. 
UNITS 
(E.  M. 
SYSTEM) 

RATIO 
E.  M. 
UNIT 
TO  E.  S. 
UNIT 

RATIO 
PRAC- 
TICAL 
UNIT  TO 
C.G.S. 
UNIT 

NAME 

Magnet  pole   .... 

F-Z*. 

r-2 

g.2  cm.  2 

sec. 

Magnetic  field  strength 

F 

fif       •*•  # 

s1 

gauss 

m 

cm.  2  sec. 

Magnetic  flux     .    .    . 

$  =  Ma  t 

g.*  cm.  5 

maxwell 

sec. 

Magnetic  moment  .    . 

M  -  ml 

g.s  cmj 

sec. 

Magnetization    .    .    . 

M      m 

g.- 

3       V       a 

cm.^  sec. 

Magnetic  induction    . 

<f> 

$  =  -=#,  +  47Tj{ 

a 

g> 

cm.  2  sec. 

Current  

F  =  IIH 

g.^  cm.  ^ 

»§ 

10-1 

ampere 

sec. 

Quantity     

Q  =  It 

g.'  cm.  ^ 

V 

10-1 

coulomb 

Potential    .    .    . 

'-? 

gj  cm.  5 

-.    i 

108 

volt 

sec.2 

Resistance  

I-PD 

"IT 

cm. 

sec. 

v--2 

109 

ohm 

Capacity     

c      Q 

sec.2 

V1 

10-9 

farad 

~  PD 

cm. 

Self-induction    .    .    . 

$>  =  LI 

cm. 

109   • 

henry 

*  For  air. 

t  For  air,  otherwise  4>  =  cBa. 


c&  =  cK  for  air,  since  cf  —  0. 
v  =  3  x  10k 


378 


ELECTRICITY,  SOUND,  AND  LIGHT 


NATURAL  SINES 


A  TIO'lP 

o 

1 

2 

3 

4 

5 

Q 

g 

g 

Complement 

XXllglC 

Difference 

0° 

0.0000 

0017 

0035 

0052 

0070 

008" 

0105 

0122 

0140 

0157 

0175 

89° 

1 

0175 

0192 

0209 

0227 

0244 

0262 

0279 

0297 

0314 

0332 

0349 

88 

2 

0349 

0366 

0384 

0401 

0419 

0436 

0454 

0471 

0488 

0506 

0523 

87 

8 

0523 

0541 

0558 

0576 

0593 

0610 

0628 

0645 

0663 

0680 

0698 

86 

4 

0698 

0715 

0732 

0750 

0767 

0785 

0802 

0819 

0837 

0854 

0872 

85 

5 

0.0872 

0889 

0906 

0924 

0941 

0958 

0976 

0993 

1011 

1028 

1045 

84 

6 

1045 

1063 

1080 

1097 

1115 

1132 

1149 

116" 

1184 

1201 

1219 

83 

7 

1219 

1236 

1253 

1271 

1288 

1305 

1323 

1340 

1357 

1374 

1392 

82 

8 

1392 

1409 

1426 

1444 

1461 

1478 

1495 

1513 

1530 

1547 

1564 

81 

9 

1564 

1582 

1599 

1616 

1633 

1650 

1668 

1685 

1702 

1719 

1736 

80 

10 

0.1736 

1754 

1771 

1788 

1805 

1822 

1840 

1857 

1874 

1891 

1908 

79 

11 

1908 

1925 

1942 

1959 

1977 

1994 

2011 

2028 

2045 

.2062 

2079 

78 

12 

2079 

2096 

2113 

2130 

2147 

2164 

2181 

2198 

2215 

2233 

2250 

77  " 

13 
14 

2250 
2419 

2267 
2436 

2284 
2453 

2300 

2470 

2317 

2487 

2334 
2504 

2351 
2521 

2368 

2538 

2385 
2554 

2403 
25J1 

^2419 

SJfcS 

76 
•75 

15 

0.2588 

2605 

2622 

2639 

2656 

2672 

2689 

2706 

2723 

2740 

2756 

74 

16 

2756 

2773 

2790 

2807 

2823 

2840 

2857 

2874 

2890 

2907 

2924 

73 

17 

2924 

2940 

2957 

2974 

2990 

3007 

3024 

3040 

3057 

3074 

3090 

72 

18 

3090 

3107 

3123 

3140 

3156 

3173 

3190 

3206 

3223 

3239 

3256 

71 

19 

3256 

3272 

3289 

3305 

3322 

3338 

3355 

3371 

3387 

3404 

3420 

70 

20 

0.3420 

3437 

3453 

3469 

3486 

3502 

3518 

3535 

3551 

3567 

3584 

69 

21 

3584 

3600 

3616 

3633 

3649 

3665 

3681 

3697 

3714 

3730 

3746 

68 

22 

3746 

3762 

3778 

3795 

3811 

3827 

3843 

3859 

3875 

3891 

3907 

67 

23 

3907 

3923 

3939 

3955 

3971 

3987 

4003 

4019 

4035 

4051 

4067 

60  18 

24 

4067 

4083 

4099 

4115 

4131 

4147 

4163 

4179 

4195 

4210 

4226 

65 

25 

0.4226 

4242 

4258 

4274 

4289 

4305 

4321 

4337 

4352 

4368 

4384 

64 

26 

4384 

4399 

4415 

4431 

4446 

4462 

4478 

4493 

4509 

4524 

4540 

63 

27 

4540 

4555 

4571 

4586 

4602 

4617 

4633 

4648 

4664 

4679 

4695 

62 

28 

4695 

4710 

4726 

4741 

4756 

4772 

4787 

4802 

4818 

4833 

4848 

61 

29 

4848 

4863 

4879 

4894 

4909 

4924 

4939 

4955 

4970 

4985 

5000 

60 

30 

0.5000 

5015 

5030 

5045 

5060 

5075 

5090 

5105 

5120 

5135 

5150 

59  15 

31 

5150 

5165 

5180 

5195 

5210 

5225 

5240 

5255 

5270 

5284 

5299 

58 

32 

5299 

5314 

5329 

5344 

5358 

5373 

5388 

5402 

5417 

5432 

5446 

57 

33 

5446 

5461 

5476 

5490 

5505 

5519 

5534 

5548 

5563 

5577 

5592 

56 

34 

5592 

5606 

5621 

5635 

5650 

5664 

5678 

5693 

5707 

5721 

5736 

55 

35 

0.5736 

5750 

5764 

5779 

5793 

5807 

5821 

5835 

5850 

5864 

5878 

54 

36 

5878 

5892 

5906 

5920 

5934 

5948 

5962 

5976 

5990 

6004 

6018 

53  i* 

37 

6018 

6032 

6046 

6060 

6074 

6088 

6101 

6115 

6129 

6143 

6157 

52 

38 

6157 

6170 

6184 

6198 

6211 

6225 

6239 

6252 

6266 

6280 

6293 

51 

39 

6293 

6307 

6320 

6334 

6347 

6361 

6374 

6388 

6401 

6414 

6428 

50 

40 

0.6428 

6441 

6455 

6468 

6481 

6494 

6508 

6521 

6534 

6547 

6561 

49 

41 

6561 

6574 

6587 

6600 

6613 

6626 

6639 

6652 

6665 

6678 

6691 

48  13 

42 

6691 

6704 

6717 

6730 

6743 

6756 

6769 

6782 

6794 

6807 

6820 

47 

43 

6820 

6833 

6845 

6858 

6871 

6884 

6896 

6909 

6921 

6934 

6947 

4<> 

44° 

6947 

6959 

6972 

6984 

6997 

7009 

7022 

7034 

7046 

7059 

7071 

45° 

Complement 

.9 

.8 

.7 

.6 

.5 

.4 

.3 

.2 

.1 

.0 

Angle 

NATURAL  COSINES 


TABLES 


379 


NATURAL  SINES 


Complement 

Angle 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

Difference 

45° 

0.7071 

7083 

7096 

7108 

7120 

7133 

7145 

7157 

71G9 

7181 

7193 

41° 

40 

7193 

7206 

7218 

7230 

7242 

7254 

7266 

7278 

7290 

7302 

7314 

43  19 

47 

7314 

7325 

7337 

7349 

7361 

7373 

7385 

7396 

7408 

7420 

7431 

42 

48 

7431 

7443 

7455 

7466 

7478 

7490 

7501 

7513 

7524 

7536 

7547 

41 

49 

7547 

7559 

7570 

7581 

7593 

7604 

7615 

7627 

7638 

7649 

7660 

40 

50 

0.7660 

7672 

7683 

7694 

7705 

7716 

7727 

7738 

7749 

7760 

7771 

C9 

51 

7771 

7782 

7793 

7804 

7815 

7826 

7837 

7848 

7859 

7869 

7880 

38  " 

52 

7880 

7891 

7902 

7912 

7923 

7934 

7944 

7955 

7965 

7976 

7986 

37 

53 

7986 

7997 

8007 

8018 

8028 

8039 

8049 

8059 

8070 

8080 

8090 

36 

54 

8090 

8100 

8111 

8121 

8131 

8141 

8151 

8161 

8171 

8181 

8192 

35 

55 

0.8192 

8202 

8211 

8221 

8231 

8241 

8251 

8261 

8271 

8281 

8290 

3410 

50 

8290 

8300 

8310 

8320 

8329 

8339 

8348 

8358 

8368 

8377 

8387 

33 

57 

8387 

8396 

8406 

8415 

8425 

8434 

8443 

8453 

8462 

8471 

.£480 

32 

58 

8480 

8490 

8499 

8508 

8517 

8526 

8536 

8545 

8554 

8563 

8572 

31 

59 

8572 

8581 

8590 

8599 

8607 

8616 

8625 

8634 

8643 

8652 

8660 

30  9 

60 

0.8660 

8669 

8678 

8686 

8695 

8704 

8712 

8721 

8729 

8738 

8746 

29 

61 

8746 

8755 

8763 

8771 

8780 

8788 

8796 

8805 

8813 

8821 

8829 

28 

62 

8829 

8838 

8846 

8854 

8862 

8870 

8878 

8886 

8894 

8902 

8910 

27  s 

63 

8910 

8918 

8926 

8934 

8942 

8949 

8957 

8965 

8973 

8980 

8988 

26 

64 

8988 

8996 

9003 

9011 

9018 

9026 

9033 

9041 

9048 

9056 

9063 

25 

65 

0.9063 

9070 

9078 

9085 

9092 

9100 

9107 

9114 

9121 

9128 

9135 

24 

66 

9135 

9143 

9150 

9157 

9164 

9171 

9178 

9184 

9191 

9198 

9205 

23  ' 

67 

9205 

9212 

9219 

9225 

9232 

9239 

9245 

9252 

9259 

9265 

9272 

22 

68 

9272 

9278 

9285 

9291 

9298 

9304 

9311 

9317 

9323 

9330 

9336 

21 

69 

9336 

9342 

9348 

9354 

9361 

9367 

9373 

9379 

9385 

9391 

9397 

20  6 

70 

0.9397 

9403 

9409 

9415 

9421 

9426 

9432 

9438 

9444 

9449 

9455 

19 

71 

9455 

9461 

9466 

9472 

9478 

9483 

9489 

9494 

9500 

9505 

9511 

18 

72 

9511 

9516 

9521 

9527 

9532 

9537 

9542 

9548 

9553 

9558 

9563 

17 

73 

9563 

9568 

9573 

9578 

9583 

9588 

9593 

9598 

9603 

9608 

9613 

16  s 

74 

9613 

9617 

9622 

9627 

9632 

9636 

9641 

9646 

9650 

9655 

9659 

15 

75 

0.9659 

9664 

9668 

9673 

9677 

9681 

9686 

9690 

9694 

9699 

9703 

14 

76 

9703 

9707 

9711 

9715 

9720 

9724 

9728 

9732 

9736 

9740 

9744 

13  « 

77 

9744 

9748 

9751 

9755 

9759 

9763 

9767 

9770 

9774 

9778 

9781 

12 

78 

9781 

9785 

9789 

9792 

9796 

9799 

9803 

9806 

9810 

9813 

9816 

11 

79 

9816 

9820 

9823 

9826 

9829 

9833 

9836 

9839 

9842 

9845 

9848 

10 

80 

0.9848 

9851 

9854 

9857 

9860 

9863 

9866 

9869 

9871 

9874 

9877 

9  3 

81 

9877 

9880 

9882 

9885 

9888 

9890 

9893 

9895 

9898 

9900 

9903 

8 

82 

9903 

9905 

9907 

9910 

9912 

9914 

9917 

9919 

9921 

9923 

9925 

7 

83 

9925 

9928 

9930 

9932 

9934 

9936 

9938 

9940 

9942 

9943 

9945 

6  2 

84 

9945 

9947 

9949 

9951 

9952 

9954 

9956 

9957 

9959 

9960 

9962 

5 

85 

0.9962 

9963 

9965 

9966 

9968 

9969 

9971 

9972 

9973 

9974 

9976 

4 

86 

9976 

9977 

9978 

9979 

9980 

9981 

9982 

9983 

9984 

9985 

9986 

3  * 

87 

9986 

9987 

9988 

9989 

9990 

9990 

9991 

9992 

9993 

9993 

9994 

2 

88 

9994 

9995 

9995 

9996 

9996 

9997 

9997 

9997 

9998 

9998 

9998 

1 

89° 

9998 

9999 

9999 

9999 

9999 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

0°<> 

Complement 

.9 

.8 

.7 

.6 

.5 

.4 

.3 

.2 

.1 

.0 

Angle 

NATURAL  COSINES 


380 


ELECTRICITY,  SOUND,  AND  LIGHT 


NATURAL  TANGENTS 


1 

Complement 

Angle 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

Difference 

0° 

0.0000 

0017 

0035 

0052 

0070 

0087 

0105 

0122 

0140 

0157 

0175 

89° 

1 

0175 

0192 

0209 

0227 

0244 

0262 

0279 

0297 

0314 

0332 

0349 

88 

2 

0349 

0367 

0384 

0402 

0419 

0437 

0454 

0472 

0489 

0507 

0524 

87 

3 

0524 

0542 

0559 

0577 

0594 

0612 

0629 

0647 

0664 

0682 

0699 

86 

4 

0699 

0717 

0734 

0752 

0769 

0787 

0805 

0822 

0840 

0857 

0875 

85 

5 

0.0875 

0892 

0910 

0928 

0945 

0963 

0981 

0998 

1016 

1033 

1051 

84 

6 

1051 

1069 

1086 

1104 

1122 

1139 

1157 

1175 

1192 

1210 

1228 

83 

7 

1228 

1246 

1263 

1281 

1299 

1317 

1334 

1352 

1370 

1388 

1405 

82 

8 

1405 

1423 

1441 

1459 

1477 

1495 

1512 

1530 

1548 

1566 

1584 

81 

9 

1584 

1602 

1620 

1638 

1655 

1673 

1691 

1709 

1727 

1745 

1763 

80 

10 

0.1763 

1781 

1799 

1817 

1835 

1853 

1871 

1890 

1908 

1926 

1944 

79  18 

11 

1944 

1962 

1980 

1998 

2016 

2035 

2053 

2071 

2089 

2107 

2126 

78 

12 

2126 

2144 

2162 

2180 

2199 

2217 

2235 

2254 

2272 

2290 

2309 

77 

13 

2309 

2327 

2345 

2364 

2382 

2401 

2419 

2438 

2456 

2475 

2493 

76 

14 

2493 

2512 

2530 

2549 

2568 

2586 

2605 

2623 

2642 

2661 

2679 

75 

15 

0.2679 

2698 

2717 

2736 

2754 

2774 

2792 

2811 

2830 

2849 

2867 

74 

16 

2867 

2886 

2905 

2924 

2943 

2962 

2981 

3000 

3019 

3038 

3057 

73  19 

17 

3057 

3076 

3096 

3115 

3134 

3153 

3172 

3191 

3211 

3230 

3249 

72 

18 

3249 

3269 

3288 

3307 

3327 

3346 

3365 

3385 

3404 

3424 

3443 

71 

19 

3443 

3463 

3482 

3502 

3522 

3541 

3561 

3581 

3600 

3620 

3640 

70 

20 

0.3640 

3659 

3679 

3699 

3719 

3739 

3759 

3779 

3799 

3819 

3839 

69 

21 

3839 

3859 

3879 

3899 

3919 

3939 

3959 

3979 

4000 

4020 

4040 

68  20 

22 

4040 

4061 

4081 

4101 

4122 

4142 

4163 

4183 

42C4 

4224 

4245 

67 

23 

4245 

4265 

4286 

4307 

4327 

4348 

4369 

4390 

4411 

4431 

4452 

66 

24 

4452 

4473 

4494 

4515 

4536 

4557 

4578 

4599 

4621 

4642 

4663 

65  21 

25 

0.4663 

4684 

4706 

4727 

4748 

4770 

4791 

4813 

4834 

4856 

4877 

64 

26 

4877 

4899 

4921 

4942 

4964 

4986 

5008 

5029 

5051 

5073 

5095 

63 

27 

5095 

5117 

5139 

5161 

5184 

5206 

5228 

5250 

5272 

5295 

5317 

62  22 

28 

5317 

5340 

5362 

5384 

5407 

5430 

5452 

5475 

5498 

5520 

5543 

61 

29 

5543 

5566 

5589 

5612 

5635 

5658 

5681 

5704 

5727 

5750 

5774 

60  23 

30 

0.5774 

5797 

5820 

5844 

5867 

5890 

5914 

5938 

5961 

5985 

6099 

59 

31 

6009 

6032 

6056 

6080 

6104 

6128 

6152 

6176 

6200 

6224 

6249 

58  2* 

32 

6249 

6273 

6297 

6322 

6346 

6371 

6395 

6420 

6445 

6469 

6494 

57 

33 

6494 

6519 

6544 

6569 

6594 

6619 

6644 

6669 

6694 

6720 

6745 

56  25 

34 

6745 

6771 

6796 

6822 

6847 

6873 

6899 

6924 

6950 

6976 

7002 

55 

35 

0.7002 

7028 

7054 

7080 

7107 

7133 

7159 

7186 

7212 

7239 

7265 

54  2S 

36 

7265 

7292 

7319 

7346 

7373 

7400 

7427 

7454 

7481 

7508 

7536 

53^ 

37 

7536 

7563 

7590 

7618 

7646 

7673 

7701 

7729 

7757 

7785 

7813 

52  23 

38 

7813 

7841 

7869 

7898 

7926 

7954 

7983 

8012 

8040 

8069 

8098 

61  28 

39 

8098 

8127 

8156 

8185 

8214 

8243 

8273 

8302 

8332 

8361 

8391 

50  29 

40 

0.8391 

8421 

8451 

8481 

8511 

8541 

8571 

8601 

8632 

8662 

8693 

49  30 

41 

8693 

8724 

8754 

8785 

8816 

8847 

8878 

8910 

8941 

8972 

9004 

48  31 

42 

9004 

9036 

9067 

9099 

9131 

9163 

9195 

9228 

9260 

9293 

9325 

47  32 

43 

9325 

9358 

9391 

9424 

9557 

9490 

9523 

9556 

9590 

9623 

9657 

46  33 

44° 

9657 

9691 

9725 

9759 

9793 

9827 

9861 

9896 

9930 

9965 

1.0000 

45034 

Complement 

.9 

.8 

.7 

.6 

.5 

.4 

.3 

.2 

.1 

.0 

Angle 

NATURAL  COTANGENTS 


TABLES 


381 


NATURAL  TANGENTS 


Angle 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

Dii. 

45° 

1.0000 

1.0035 

1.0070 

1.0105 

1.0141 

1.0176 

1.0212 

1.0247 

1.0283 

1.0319 

36 

46 

1.0355 

1.0392 

1.0428 

1.0464 

1.0501 

1.0538 

1.0575 

1.0612 

1.0649 

1.0686 

37 

47 

1.0724 

1.0761 

1.0799 

1.0837 

1.0875 

1.0913 

1.0951 

1.0990 

1.1028 

1.1067 

38 

48 

1.1106 

1.1145 

1.1184 

1.1224 

1.1263 

1.1303 

1.1343 

1.1383 

1.  1423 

1.1463 

40 

49 

1.1504 

1.1544 

1.1585 

1.1626 

1.1667 

1.1708 

1.1750 

1.1792 

1.1833 

1.1875 

41 

50 

1.1918 

1.1960 

1.2002 

1.2045 

1.2088 

1.2131 

1.2174 

1.2218 

1.2261 

1.2305 

43 

51 

1.2349 

1.2393 

1.2437 

1.2482 

1.2527 

1.2572 

1.2617 

1.2662 

1.2708 

1.2753 

45 

52 

1.2799 

1.2846 

1.2892 

1.2938 

1.2985 

1.3032 

1.3079 

1.3127 

1.3175 

1.3222 

47 

53 

1.3270 

1.3319 

1.3367 

1.3416 

1.3465 

1.3514 

1.3564 

1.3613 

1.3663 

1.3713 

49 

54 

1.3764 

1.3814 

1.3865 

1.3916 

1.3968 

1.4019 

1.4071 

1.4124 

1.4176 

1.4229 

52 

55 

1.4281 

1.4335 

1.4388 

1.4442 

1.4496 

1.4550 

1.4605 

1.4659 

1.4715 

1.4770 

54 

56 

1.4826 

1.4882 

1.4938 

1.4994 

1.5051 

1.5108 

1.5166 

1.5224 

1.5282 

1.5340 

07 

57 

1.5399 

1.5458 

1.5517 

1.5577 

1.5637 

1.5697 

1.5757 

1.5818 

1.5880 

1.5941 

60 

58 

1.6003 

1.6066 

1.6128 

1.6191 

1.6255 

1.6319 

1.6383 

1.6447 

1.6512 

1.6577 

64 

59 

1.6643 

1.6709 

1.6775 

1.6842 

1.6909 

1.6977 

1.7045 

1.7113 

1.7182 

1.7251 

68 

60 

1.7321 

1.7391 

1.7461 

1.7532 

1.7603 

1.7675 

1.7747 

1.7820 

1.7893 

1.7966 

72 

61 

1.8040 

1.8115 

1.8190 

1.8265 

1.8341 

1.8418 

1.8495 

1.8572 

1.8650 

1.8728 

77 

62 

1.8807 

1.8887 

1.8967 

1.9047 

1.9128 

1.9210 

1.9292 

1.9375 

1.9458 

1.9542 

82 

63 

1.9626 

1.9711 

1.9797 

1.9883 

1.99702.0057 

2.0145 

2.0233 

2.0323 

2.0413 

88 

64 

2.0503 

2.0594 

2.0686 

2.0778 

2.0872 

2.0965 

2.1060 

2.1155 

2.1251 

2.134S 

94 

65 

2.145 

2.154 

2.164 

2.174 

2.184 

2.194 

2.204 

2.215 

2.225 

2.236 

10 

66 

2.246 

2.257 

2.267 

2.278 

2.289 

2.300 

2.311 

2.322 

2.333 

2.344 

11 

67 

2.356 

2.367 

2.379 

2.391 

2.402 

2.414 

2.426 

2.438 

2.450 

2.463 

12 

68 

2.475 

2.488 

2.500 

2.513 

2.526 

2.539 

2.552 

2.565 

2.578 

2.592 

13 

69 

2.605 

2.619 

2.633 

2.646 

2.660 

2.675 

2.689- 

2.703 

2.718 

2.733 

14 

70 

2.747 

2.762 

2.778 

2.793 

2.808 

2.824 

2.840 

2.856 

2.872 

2.888 

16 

71 

2.904 

2.921 

2.937 

2.954 

2.971 

2.989 

3.006 

3.024 

3.042 

3.060 

17 

72 

3.078 

3.096 

3.115 

3.133 

3.152 

3.172 

3.191 

3.211 

3.230 

3.250 

19 

73 

3.271 

3.291 

3.312 

3.333 

3.354 

3.376 

3.398 

3.420 

3.442 

3.465 

22 

74 

3.487 

3.511 

3.534 

3.558 

3.582 

3.606 

3.G30 

3.655 

3.681 

3.700 

25 

75 

3.732 

3.758 

3.785 

3.812 

3.839 

3.867 

3.895 

3.923 

3.952 

3.981 

28 

76 

4.011 

4.041 

4.071 

4.102 

4.134 

4.165 

4.198 

4.230 

4.264 

4.297 

32 

77 

4.331 

4.366 

4.402 

4.437 

4.474 

4.511 

4.548 

4.586 

4.625 

4.665 

37 

78 

4.705 

4.745 

4.787 

4.829 

4.872 

4.915 

4.959 

5.005 

5.050 

5.097 

44 

79 

5.145 

5.193 

5.242 

5.292 

5.343 

5.396 

5.449 

5.503 

5.558 

5.614 

52 

80 

5.67 

5.73 

5.79 

5.85 

5.91 

5.98 

6.04 

6.11 

6.17 

6.24 

7 

81 

6.31 

6.39 

6.46 

6.54 

6.61 

6.G9 

6.77 

6.85 

6.94 

7.03 

8 

82 

7.12 

7.21 

7.30 

7.40 

7.49 

7.60 

7.70 

7.81 

7.92 

8.03 

10 

83 

8.14 

8.26 

8.39 

8.51 

8.64 

8.78 

8.92 

9.06 

9.21 

9.36 

li 

84 

9.51 

9.68 

9.84 

10.0 

10.2 

10.4 

10.6 

10.8 

11.0 

11.2 

85 

11.4 

11.7 

11.9 

12.2 

12.4 

12.7 

13.0 

13.3 

13.6 

14.0 

3 

86 

14.3 

14.7 

15.1 

15.5 

15.9 

16.3 

16.8 

17.3 

17.9 

18.5 

6 

87 

19.1 

19.7 

20.4 

21.2 

22.0 

22.9 

23.9 

24.9 

26.0 

27.3 

88 

28.6 

30.1 

31.8 

33.7 

35.8 

38.2 

40.9 

44.1 

47.7 

52.1 

89° 

57. 

64. 

72.   82. 

95. 

115. 

143. 

191. 

286. 

573. 

Angle 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

NATURAL  TANGENTS 


382 


ELECTRICITY,  SOUND,  AND   LIGHT 


LOGARITHMS 


10 

0 

0000 

1 
0043 

2 

0086 

3 

4 

0170 

5 

6 

7 

0294 

8 

9 

1  2  3 

450 

789 

0128 

0212 

0253 

0334 

0374 

4  8  12 

17  21  25 

29  33  37 

11 

12 
13 

0414 
0792 
1139 

0453 

0828 
1173 

0492 
0864 
1206 

0531 
0899 
1239 

0569 
0934 

1271 

0607 
0969 
1303 

0645 
1004 
1335 

0682 
1038 
1367 

0719 
1072 
1399 

0755 
1106 
1430 

4  8  11 
3  7  10 
3  6  10 

15  19  23 
14  17  21 
13  16  19 

26  30  34 
24  28  31 
23  26  29 

14 
15 
16 

1461 
1761 
2041 

1492 

1790 
2068 

1523 

1818 
2095 

1553 
1847 
2122 

1584 
1875 

2148 

1614 
1903 
2175 

1644 
1931 
2201 

1673 
1959 

2227 

1703 

1987 
2253 

1732 
2014 
2279 

369 

368 
358 

12  15  18 
11  14  17 
11  13  16 

21  24  27 
20  22  25 
18  21  24 

17 

18 
19 

2304 
2353 

2788 

2330 
2577 
2810 

2355 
2601 
2833 

2380 
2625 
2856 

2405 
2648 

2878 

2430 
2672 
2900 

2455 
2695 
2923 

2480 
2718 
2945 

2504 

2742 
2967 

2529 
2765 
2989 

257 

257 
247 

10  12  15 
9  12  14 
9  11  13 

17  20  22 
16  19  21 
16  18  20 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

246 

8  11  13 

15  17  19 

21 
22 
23 

3222 
3424 
3617 

3243 
3444 
3636 

3263 
3464 
3655 

3284 
3483 
3674 

3304 
3502 
3692 

3324 
3522 
3711 

3345 
3541 
3729 

3365 

3560 
3747 

3385 
3579 
3766 

3404 
3598 

3784 

246 
246 
246 

8  10  12 
8  10  12 
7  9  11 

14  16  18 
14  15  17 
13  15  17 

24 
25 
26 

3802 
3979 
4150 

3820 
3997 
4166 

3838 
4014 
4183 

3856 
4031 
4200 

3874 
4048 
4216 

3892 
4065 
4232 

3909 

4082 
4249 

3927 
4099 
4265 

3945 
4116 
4281 

8962 
4133 
4298 

245 
235 
235 

7  9  11 
7  9  10 
7  8  10 

12  14  16 
12  14  15 
11  13  15 

27 

28 
29 

4314 
4472 
4624 

4330 
4487 
4639 

4346 
4502 
4654 

4362 

4518 
4669 

4378 
4533 
4683 

4393 

4548 
4698 

4409 
4564 
4713 

4425 
4579 

4728 

4440 
4594 
4742 

4456 
4609 

4757 

235 
235 
1  3  4 

689 
689 
679 

11  13  14 
11  12  14 
10  12  13 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

1  3  4 

679 

10  11  13 

31 
32 
33 

4914 
5051 
5185 

4928 
5065 
5198 

4942 
5079 
5211 

4955 
5092 
5224 

4969 
5X05 
5237 

4983 
5119 
5250 

4997 
5132 
5263 

5011 
5145 
5276 

5024 
5159 
5289 

5038 
5172 
5302 

1  3  4 
1  3  4 
1  3  4 

678 
578 
568 

10  11  12 
9  11  12 
9  10  12 

34 
35 
36 

5315 
5441 
5563 

5328 
5453 
5575 

5340 
5465 

5587 

5353 

5478 
5599 

5366 

5490 
5611 

5378 
5502 
5623 

5391 
5514 
5635 

5403 

5527 
5647 

5416 
5539 
5658 

5428 
5551 
5670 

1  3  4 

1  2  4 
1  2  4 

568 
567 
567 

9  10  11 
9  10  11 
8  10  11 

37 
38 
39 

5682 
5798 
5911 

5694 
5809 
5922 

5705 
5821 
5933 

5717 
5832 
5944 

5729 
5843 
5955 

5740 

5855 
5966 

5752 
5866 
5977 

5763 

5877 
5988 

5775 
5888 
5999 

5786 
5899 
6010 

1  2  3 
1  2  3 
1  2  3 

567 
567 
457 

8  9  10 
8  9  10 
8  9  10 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

1  2  3 

456 

8  9  10 

41 
42 
43 

6128 
6232 
6335 

6138 
6243 
6345 

6149 
6253 
6355 

6160 
6263 
6365 

6170 
6274 
6375 

6180 
6284 
6385 

6191 
6294 
6395 

6201 
6304 
6405 

6212 
6314 
6415 

6222 
6325 
6425 

123 
1  2  3 
1  2  3 

456 
456 
4  5  G 

789 

789 
789 

44 
45 
46 

6435 
6532 

6628 

6444 
6542 
6637 

6454 
6551 
6646 

6464 
6561 
6656 

6474 
6571 
6665 

6484 
6580 
6675 

6493 

6590 
6684 

6503 
6599 
6693 

6513 
6609 
6702 

6522 
6618 
6712 

123 
123 
1  2  3 

456 
456 
456 

789 
789 

778 

47 

48 
49 

6721 
6812 
6902 

6730 
6821 
6911 

6739 
6830 
6920 

6749 
6839 
6928 

6758 
6848 
6937 

6767 

6857 
6946 

6776 
6866 
6955 

6785 
6875 
6964 

6794 

6884 
6972 

6803 
6893 
6981 

1  2  3 
1  2  3 
1  2  3 

455 
445 
445 

678 
678 
6  7,  8 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

1  2  3 

345 

678 

51 
52 
53 

7076 
7160 
7243 

7084 
7168 
7251 

7093 

7177 
7259 

7101 

7185 

7267 

7110 
7193 

7275 

7118 

7202 
7284 

7126 
7210 
7292 

7135 

7218 
7300 

7143 
7226 

7308 

7152 
7235 
7316 

1  2  3 

1  2  2 
1  2  2 

345 
345 
345 

678 
677 
667 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

1  2  2 

345 

667 

TABLES 


383 


LOGARITHMS 


55 

0 

l 

2 

3 

4 

5 

6 

7 

8 

9 

1  2  3 

456 

789 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

122 

345 

567 

56 
57 

58 

7482 
7559 
7634 

7490 
7566 
7642 

7497 
7574 
7649 

7505 
7582 
7657 

7513 
7589 
7664 

7520 
7597 
7672 

7528 
7604 
7679 

7536 
7612 
7686 

7543 
7619 
7694 

7551 
7627 
7701 

122 
122 
1  1  2 

345 
345 
344 

567 
567 
567 

59 
60 
61 

7709 

7782 
7853 

7716 

7789 
7860 

7723 

7796 
7868 

7731 

7803 
7875 

7738 
7810 

7882 

7745 

7818 
7889 

7752 
7825 
7896 

7760 

7832 
7903 

7767 
7839 
7910 

7774 
7846 
7917 

112 
112 
112 

344 
344 
344 

567 
566 
566 

62 
63 
64 

7924 
7993 
8062 

7931 
8000 
8069 

7938 
8007 
8075 

7945 
8014 

8082 

7952 
8021 
8089 

7959 
8028 
8096 

7966 
8035 
8102 

7973 
8041 
8109 

7980 
8048 
8116 

7987 
8055 
8122 

112 
112 
112 

334 
334 
334 

566 
556 
556 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

112 

334 

556 

66 
67 

68 

8195 
8261 
8325 

8202 
8267 
8331 

8209 
8274 
8338 

8215 
8280 
8344 

8222 
8287 
8351 

8228 
8293 
8357 

8235 
8299 
8363 

8241 
8306 
8370 

8248 
8312 
8376 

8254 
8319 
8382 

1  1 
1  1 
1  1 

334 
334 
334 

556 
556 
456 

69 
70 
71 

8388 
8451 
8513 

8395 
8457 
8519 

8401 
8463 
8525 

8407 
8470 
8531 

8414 
8476 
8537 

8420 

8482 
8543 

8426 
8488 
8549 

8432 
8494 
8555 

8439 
8500 
8561 

8445 
8506 
8567 

1  1 
1  1 
1  1 

234 
234 
234 

456 
456 
455 

72 
73 
74 

8573 
8633 
8692 

8579 
8639 
8698 

8585 
8645 
8704 

8591 
8651 
8710 

8597 
8657 
8716 

8603 
8663 

8722 

8609 
8669 

8727 

8615 
8675 
8733 

8621 

8681 
8739 

8627 
8686 
8745 

1  1 
1  1 
1  1 

2  3 
2  3 
2  3 

455 
455 
455 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

1  1 

2  3 

455 

76 

77 
78 

8808 
8865 
8921 

8814 
8871 
8927 

8820 
8876 
8932 

8825 
8882 
8938 

8831 
8887 
8943 

8837 
8893 
8949 

8842 
8899 
8954 

8848 
8904 
8960 

8854 
8910 
8965 

8859 
8915 
8971 

1  1 
1  1 
1  1 

2  3 
2  3 
2  3 

455 
445 
445 

79 

80 
81 

8976 
9031 
9085 

8982 
9036 
9090 

8987 
9042 
9096 

8993 
9047 
9101 

8998 
9053 
9106 

9004 
9058 
9112 

9009 
90G3 
9117 

9015 
9069 
9122 

9020 
9074 
9128 

9025 
9079 
9133 

1  1  2 
1  1  2 
112 

2  3 

2  3 

2  3 

445 
445 
445 

82 
83 

84 

9138 
9191 
9243 

9143 
9196 

9248 

9149 
9201 
9253 

9154 
9206 
9258 

9159 
9212 
9263 

9165 
9217 
9269 

9170 
9222 
9274 

9175 
9227 
9279 

9180 
9232 
9284 

9186 
9238 
9289 

1  1  2 
1  1  2 
112 

2  3 
2  3 
2  3 

445 
445 
445 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

1  1  2 

2  3 

445 

86 

87 
88 

9345 
9395 
9445 

9350 
9400 
9450 

9355 
9405 
9455 

9360 
9410 
9460 

9365 
9415 
9465 

9370 
9420 
9469 

9375 
9425 
9474 

9380 
9430 
9479 

9385 
9435 

9484 

9390 
9440 
9489 

1  1  2 
0  1  1 
0  1  1 

2  3 
2  2 
2  2 

445 
344 
344 

89 
90 
91 

9494 
9542 
9590 

9499 
9547 
9595 

9504 
9552 
9600 

9509 
9557 
9605 

9513 
9562 
9609 

9518 
9566 
9614 

9523 
9571 
9619 

9528 
9576 
9624 

9533 
9581 
9628 

9538 
9586 
9633 

0  1  1 
0  1  1 
0  1  1 

2  2 
2  2 
2  2 

344 
344 
344 

92 
93 
94 

9638 
9685 
9731 

9643 
9689 
9736 

9647 
9694 
9741 

9652 
9699 
9745 

9657 
9703 
9750 

9661 
9708 
9754 

9666 
9713 
9759 

9671 
9717 
9763 

9675 
9722 
9768 

9680 
9727 
9773 

0  1  1 
0  1  1 
0  1  1 

2  2 
2  2 

2  2 

344 
344 
344 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

0  1  1 

2  2 

344 

96 
97 
98 

9823 
9868 
9912 

9827 
9872 
9917 

9832 

9877 
9921 

9836 
9881 
9926 

9841 
9886 
9930 

9845 
9890 
9934 

9850 
9894 
9939 

9854 
9899 
9943 

9859 
9903 
9948 

9863 
9908 
9952 

0  1  1 
0  1  1 
0  1  1 

2  2 

2  2 
2  2 

344 
344 
344 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

0  1  1 

2  2 

334 

INDEX 


(Numbers  refer  to  pages) 


Aberration,  spherical,  288 

chromatic,  305 
Achromatic  lens,  305 
Adiabatic  compression,  193 
Alpha  rays,  341 
Amagat,  190 
Ammeter,  39,  84 
Ampere,  35 
Ampere,  unit,  35 
Amplitude,  defined,  223 
Arago,  331 

Arc-light  efficiency,  299 
Arrhenius,  181 
Atom,  nature  of,  336,  345 
Atomic  weights,  table  of,  370 

cB  =  magnetic  induction,  160 
o/3  and  SK  curve,  161,  169 
Ballistic  galvanometer,  97 
Ballistic   method,    for  magnetization, 
168 

for  comparing  E.M.F.'s,  122 
Barometric  redactions,  table  of,  375 
Bartholinus,  321 
Battery,  Daniell,  117 

E.M.F.  of,  113,  117 

resistance  of,  114,  117 

polarization  of,  121 

Leclanche',  124 

Clark  standard,  121 
Beats,  234 
Becquerel,  340 
Beta  rays,  341 
Biot,  332 

Boyle's  law,  43,  192,  194 
Brewster's  law,  318 
Bright-line  spectra,  307 

Candle,  British  standard,  294 
Capacity,  denned,  90 

units  of,  97 

measurement  of,  93 

laws  of  combination,  103 

comparison  of,  100 

standard,  105 

calculation  of,  107 


Carcel  lamp,  294 

Cathode  rays,  333,  334 

Caustics,  286 

Charles's  law,  43 

Chromatic  aberration,  305 

Circular  polarization,  327 

Clausius,  177 

Clausius's  dissociation  theory,  177 

Coercive  force,  165 

Colladon,  190 

Colors  produced  by  polarized  light  in 

crystals,  328 
Compressional  waves,  in  fluids,  187 

in  rods,  213 

Condensation,  reflection  of,  195 
Condenser,  capacity  of,  91 

formula  for,  107 

mica,  109 
Conduction,  electrical,  3,  24 

electrolytic,  176 
Conductivity,  molecular,  180 

specific,  183 

tables  of,  see  Resistance 
Continuous  spectrum,  306 
Cooper-Hewitt  lamp,  299 
Corpuscular  theory  of  light,  238 
Cosines,  tables  of  natural,  378,  379 
Coulomb,  35 
Coulomb,  unit,  35 

Critical  angle  of  total  reflection,  288 
Crookes,  334,  344 
Curie,  M.  and  Mme.,  343 
Current,  electrical,  defined,  28 

magnetic  effect  of,  29 

direction  of,  31 

unit  of,  33,  35,  77 

measuring  instruments,  86 
Curvature,  defined,  271 

Damping  of  galvanometer,  65 

damping  factor,  95 
Daniell  battery,  117 
Dark-line  spectra,  307 
D'Arsonval  galvanometer,  63,  80,  97 
Defining  equations,  re'sume'  of,  377 
Demagnetization,  method  of,  170 


385 


386 


ELECTRICITY,  SOUND,  AND  LIGHT 


Density,  tables  of: 

air,  368 

gases,  373 

mercury,  368 

solids,  373 

liquids,  369,  373 

water,  368 
Dextrose,  332 

Diamagnetic  substances,  159 
Diatonic  scale,  208 
Dielectric  absorption,  103 
Dielectric  constant,  102-110 

tables  of,  372 
Diffraction,  238 

through  apertures,  242,  266 
Diffraction  grating,  258  et  seq. 
Direct-vision  spectroscope,  305 
Discontinuous  spectra,  306 
Dispersion,  Chapter  XXVI 
Dispersive  power  of  grating,  263 
Displacement  law  of  radiation,  298 
Dissociation,  electrolytic,  177 
Distribution  of  magnetism,  36,  137 
Doppler's  principle,  309 
Double  refraction,  323 
Drude,  6 
Dynamo,  the  ideal,  131 

construction  of,  134 

rule,  128 
Dynamometer,  88 

E.M.F.  =  electromotive  force,  113 
Earth  inductor,  142 
Earth,  magnetic  constants  of,  141 
Electrical  conduction,  3,  24 
Electricity,  quantity  of,  2,  35 

theory  of,  2 
electro-chemical  equivalents,  38 

table  of,  371 
Electrolysis,  37 
Electrolytic  conduction,  176 
Electro-magnetic,  induction,  126 

system  of  units,  34 

units,  table  of,  377 
Electromotive  force,  denned,  113 

induced,  127 

comparison  of,  122,  123 
Electron,  6,  337 
Electrostatic,  induction,  4 

system  of  units,  33 

field  intensity,  107 
Elliptical  polarization,  328 
Emanation,  radium,  346 
Equipotential  surfaces,  11 

determination  of,  13 
Even  temperament,  208 
Extraordinary  ray,  322 


Farad,  97 

Faraday,  37,  79,  97,  176 
laws  of  electrolysis,  37 
concept  of  lines  of  force,  79 

Fields  of  force,  8 

Figure  of  merit  of  galvanometer,  82 

Fleming's  method  for  measuring  self- 
induction,  156 

Fluorescence  due  to  radiations,  333, 344 

Focal  lengths,  273 

Franklin,  6 

Fraunhofer  lines,  307 

Fundamental  note,  201 

Galvanic  cell,  30,  117  et  seq. 
Galvanometer,  D' Arson val,  63,  80 

tangent,  35 

ballistic,  97 

Thomson,  63,  86 

direct-reading,  84 

resistance  of,  84 

constant  of,  77 

damping  of,  95 
Gamma  rays,  344 
Gases,  ionization  of,  338 
Gauss,  unit,  9 

eyepiece,  278 
Gay-Lussac's  law,  194 
Gouy,  311 

Gram-molecule,  179 
Graphical  transformation  of  cB  and  &C 

curve,  167 
Grotthus  chain,  177 

H  =  horizontal  component  of  earth's 
field,  by  magnetometer,  Chapter  II 

by  earth  inductor,  142 
3C  =  magnetic  field  intensity,  8,  128 
Heating  effect  of  a  current,  49 
Hefner  lamp,  294 
Helmholtz,  112 
Henry,  127,  149 
Henry  unit,  149 
Buy  gens,  238,  321 
Hysteresis,  104 

loop,  165 

3  =  magnetization,  161 
3  and  &C  curve,  169 
Iceland  spar,  321 
Illumination,  laws  of,  292 
Images,  formation  of,  245  et  seq. 
Incandescent  lamp  efficiency,  299 
Index  of  refraction,  274 

tables  of,  374 

by  minimum  deviation,  277 

by  total  reflection,  289 


INDEX 


387 


Induction,  coil,  152 

electro-magnetic,  126 

electrostatic,  4 

magnetic,  159 
Intensity,  of  current,  33,  35,  77 

of  electrostatic  field,  107 

of  magnetic  field,  17,  31,  150 
Interference  of  two  wave  trains,  240 
Internal    resistance  of    battery,    114, 

117 

Ionic  speeds,  182 
lonization,  of  gases,  338 

of  liquids,  Chapter  XVI 
Irrationality  of  dispersion,  304 
Isothermal  compression,  193 

Just  temperament,  210 

Kelvin,  1 
Kohlrausch,  180 
Kundt's  tube,  217 

La  Place,  193 
Lenard,  335,  336 
Lens,  formulas,  273 

images  formed  by,  245 
Lenz's  law,  127,  130 
Levulose,  332 
Light,  two  theories  of,  238 

velocity,  35,  105 
Light  waves,  238  et  seq. 
Liquids,  ionization  of,  Chapter  XVI 
Logarithms,  tables  of,  382,  383 
Longitudinal  waves,  187  et  seq.,  213  et 

seq. 

Loops,  215 
Luminescence,  298 
Lummer-Brodhun  photometer,  294 

Major  chord,  208 
Magnet,  1 

pole  strength,  1,  17 
Magnetic  creeping,  171 
Magnetic  dip,  141 
Magnetic  field  about  current,  31 
Magnetic  field  intensity,    17  et  seq., 

159 

Magnetic  moment,  18 
Magnetic  variation,  141 
Magnetism,  quantity  of,  1,  17,  136 

distribution  of,  136 
Magnetization,  defined,  161 

three  stages  of,  161 

curve  of,  168-1 70 
Magnetometer,  Experiment  2 
Mains,  316 
Melde's  experiment,  229 


Michelson,  265 

Minimum  deviation,  angle  of,  276 

measurement  of,  280 
Mirror  formulas,  273 

images  formed  by,  245  et  seq. ,  287 
Molecular  conductivity,  180 

measurement  of,  183 
Molecular  theory  of  magnetism,  134 
Moment  of  inertia  of  bar  magnet,  22 
Motor  rule,  78 
Musical  scale,  208 

Natural  period,  of  pipes,  203 

of  rods,  214 
Nernst  lamp,  299 

Newton's    deduction    of    velocity    of 
sound,  192 

experiment  on  dispersion,  300 

corpuscular  theory  of  light,  238 
Nicol  prism,  320-325 
Nodes,  215 
Normal  solution,  179 

densities  of,  369 
Norrenberg  polariscope,  316 

Oersted's  experiment,  31 
Ohm,  58 
Ohm,  unit,  59 
Ohm's  law,  58 

in  electrolytes,  178 
One-fluid  theory,  6 

Optical  efficiency  of  light  sources,  298 
Ordinary  ray,  322 
Organ  pipes,  205 
Osmotic  phenomena,  179 
Overtones,  201 

P.D.  =  potential  difference,  46-51 
Parallel  resistances,  60 

capacities,  103 

Paramagnetic  substances,  159 
Permeability,  171 
Perrin,  336 
Phase,  191 

Photometric  standards,  204 
Photometry,  Chapter  XXV 
Pipes,  open,  201 

closed,  201 

notes  to  which  p-'pes  respond,  201 

natural  periods  of,  203 
Pitchblende,  343 
Platinum  black,  185 
Polarization,  by  reflection,  316 

by  double  refracting  crystals,  321 
Polarization,  electrolytic,  116 
Polarizfd  light,  Chapter  XXVII 
Po-st-office  box,  G7-70 


388 


ELECTEICITY,  SOUND,  AND  LIGHT 


Potential,  electrical,  11 

gravitational,  9 

magnetic,  10 

of  conductor  in   electrical   equilib- 
rium, 12 

Potential  difference,  46-51 
Potentiometer   method  of  comparing 

E.M.F.'s,  122 
Practical  units,  35,  51,  59,  97,  149 

table  of,  377 

Prevost's  law  of  exchanges,  297 
Prism,  measurement  of  angle  of,  278 

Quarter-wave  plate,  326 

Quartz,  right-  and  left-handed,  332 

Radiation,  types  of,  333 
Radio-activity,  Chapter  XXVIII 

discovery  of,  340 

an  atomic  property,  345 

theory  of,  345 
Radium,  344 

Radium  A,  B,  C,  D,  E,  F,  349 
Rarefaction,  reflection  of,  196 
Ratio  of  E.M.  and  E.S.  units,  35,  105 

of  specific  heats,  193 
Rayleigh,  192,  198,  311 
Real  image,  246 
Reflection,  of  wave  trains,  195 

law  of,  247 

total,  88,  285 
Reflection  grating,  267 
Refraction,  of  light,  270 

through  prism,  275 
Residual  magnetism,  165 
Resistance,  58 

laws  of,  60 

specific,  71 

tables  of,  371,  372 
Resolving  power,  of  grating,  265 

of  lens,  251 

Resonance  of  air  columns,  196 
Retentivity,  165 
Reversal  of  spectral  lines,  309 
Riecke,  6 

Right-hand  screw  rule,  32 
Rods,  waves  in,  213  et  seq.,  233 
Rotary  polarization,  330 
Rowland,  29,  267 

Saturation,  magnetic,  135 
Scale,  musical,  208 
Self-induction,  nature  of,  146,  151 

coefficient  of,  148 

measurement  of,  149 

unit  of,  148 

illustrations  of,  150 


Sensibility  of  galvanometer,  83 
Sensitive  flame,  252 
Series,  resistances  in,  60 

capacities  in,  104 
Sines,  tables  of,  378,  379 
Sodium  lines,  reversal  of,  309 
Solar  spectrum,  307 
Solenoid,  field  strength  within,  166 
Solid  angle,  108 
Sound,  187  et  seq. 

velocity  of,  in  air,  Chapter  XVII 

velocity  of,  in  water,  190 
Specific  conductivity,  183 
Specific  resistance,  71 
Specific  resistance,  table  of,  371,  372 
Specific  heats,  table  of,  373 
Spectrometer,  277 
Spectro-photometry,  297 
Spectroscopic  analysis,  311 
Spectrum,  Chapter  XXV 

pure,  303 

grating,  262 

normal,  262 

prismatic,  304 

solar,  307 

Spinthariscope,  344 
Stationary  waves,  226 
Stefan's  law  of  radiation,  297 
Strings,  waves  in,  222  et  seq. 
Sturm,  190 
Sugar,  rotation  of  plane  of  polarization 

by,  332 
Susceptibility,  171 

Tangent  galvanometer,  35 
Tangents,  table  of  natural,  380,  381 
Temperament,  scale  of  even,  210 

scale  of  just,  209 

Temperature  coefficient  of  resistance, 
71 

tables  of,  371 
Test  coil,  142,  158 
Thermo-electro motive  forces,  113 
Thomson,  7,  336,  343,  350 
Thorium,  346 
Total  reflection,  Chapter  XXIV 

prism,  289 
Transformer,  alternating  current, 

153 
Transverse  wave  motion,  224  et  seq., 

Chapter  XXVII 
Tube  of  force,  9 
Tuning  fork,  234 
Two-fluid  theory,  5 

Units,  table  of,  377 

Uranium,  radio-activity  of,  341 


IXDEX 


389 


Uranium  X,  346 

Useful  numbers,  table  of,  375 

V=  vertical  component  of  earth's  field, 

141 

measurement  of,  by  means  of  an 
earth  inductor,  142 

Van  '«  Hoff,  179 

Vapor  pressures,  table  of,  367 

Variation,  magnetic,  141 

Velocity  of  compressional  waves,  187, 

213 
of  transverse  waves,  230 

Velocity  of  light,  35 

in  two  media  compared,  271 
of  ordinary  and  extraordinary  ray 
compared,  323  et  seq. 

Velocity  of  sound,  190,  194 
effect  of  temperature  on,  194 
independent  of  pressure,  194 
in  two  solids  compared,  215 
in  two  gases,  219 

Virtual  image,  246 

Viscosity,  table  of,  369 

Volt,  51 

Voltameter,  39 

Voltmeter,  51 
electrostatic,  352 


Water-vapor  pressures,  table  of,  367 
Wave,  front,  239 

construction  of,  240 

length,  224 

lengths  of  light  in  air  (tables),  374 

motion,  222  et  seq. 

length  by  grating,  261 

length  by  Kundt's  tube,  217 
Waves,  train  of,  190 

reflection  of,  194 
Welsbach  lamp,  299 
Wheatstone  bridge,  61  et  seq. 

measurement  of  self-induction  by, 
149 

measurement  of  molecular  conduc- 
tivity by,  184 

Wind  instruments,  musical,  206 
Wire,  sizes  and  resistances,  table  of,  376 
Wollaston,  307 

X  rays,  333,  339 

Young,  238 

Young's  modulus,  tables  of,  372 

Zeeman,  336 

Zeeman  effect,  336,  337 


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A   FIRST   COURSE   IN    PHYSICS 

By  ROBERT  A.  MILLIKAN,  Assistant  Professor  of  Physics  in 

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